Probabilistic exact universal quantum circuits for transforming unitary operations
Marco Túlio Quintino, Qingxiuxiong Dong, Atsushi Shimbo, Akihito Soeda, Mio Murao
PProbabilistic exact universal quantum circuits for transforming unitary operations
Marco Túlio Quintino, Qingxiuxiong Dong, Atsushi Shimbo, Akihito Soeda, and Mio Murao
Department of Physics, Graduate School of Science, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan (Dated: 15th of April 2020)This paper addresses the problem of designing universal quantum circuits to transform k uses ofa d -dimensional unitary input-operation into a unitary output-operation in a probabilistic heraldedmanner. Three classes of protocols are considered, parallel circuits, where the input-operations can beperformed simultaneously, adaptive circuits, where sequential uses of the input-operations are allowed,and general protocols, where the use of the input-operations may be performed without a definitecausal order. For these three classes, we develop a systematic semidefinite programming approachthat finds a circuit which obtains the desired transformation with the maximal success probability.We then analyse in detail three particular transformations; unitary transposition, unitary complexconjugation, and unitary inversion. For unitary transposition and unitary inverse, we prove that forany fixed dimension d , adaptive circuits have an exponential improvement in terms of uses k whencompared to parallel ones. For unitary complex conjugation and unitary inversion we prove that if thenumber of uses k is strictly smaller than d −
1, the probability of success is necessarily zero. We alsodiscuss the advantage of indefinite causal order protocols over causal ones and introduce the conceptof delayed input-state quantum circuits.
I. INTRODUCTION
In quantum mechanics, deterministic transformationsbetween states are represented by quantum channelsand probabilistic transformations by quantum instru-ments, which consist of quantum channels followed bya quantum measurement. Understanding the proper-ties of quantum channels and quantum instruments is astandard and well established field of research with dir-ect impact for theoretical and applied quantum physics[1, 2]. Similarly to states, quantum channels may alsobe subjected to universal transformation in a paradigmusually referred as higher order transformations. Higherorder transformations can be formalised by quantum su-permaps [3, 4] and physically implemented by meansof quantum circuits (see Fig. 1). Despite its fundamentalvalue and potential for applications ( e.g. , quantum circuitdesigning [3], quantum process tomography [5], testingcausal hypothesis [6], channel discrimination [7], align-ing reference frames [8, 9], analysing the role of causalorder [10, 11]), higher order transformations are still notwell understood when compared to quantum channelsand quantum instruments.Reversible operations play an important role in math-ematics and in various physical theories such as quantummechanics and thermodynamics. In quantum mechanics,reversible operations are represented by unitary operat-ors [12, 13]. This work considers universal transforma-tions between reversible quantum transformations, thatis, we seek for quantum circuits which implement thedesired transformation for any unitary operation of somefixed dimension without any further specific details ofthe input unitary operation. From a practical perspective,this universal requirement ensures that the circuit doesnot require any readjustments or modification when dif-ferent inputs are considered and the circuit implementsthe desired transformation even when the descriptionof the d -dimensional reversible operation is unknown. Figure 1. Pictorial representation of parallel and adaptivequantum circuits that transform k uses of a d -dimensional arbit-rary unitary input-operation described by U d into another unit-ary operation described by f ( U d ) . The circuit elements (cid:101) E and (cid:101) E i are quantum deterministic operations, i.e. , quantum channels,that may be interpreted as encoders and the element (cid:101) D standsfor decoder, a probabilistic quantum operation (quantum in-strument), that is, a quantum channel with a quantum measure-ment that when the “correct” outcome is obtained, the targettransformation is obtained perfectly. Note that the universal requirement also imposes strongconstraints on transformations which can be physicallyrealised. A well-known example which pinpoints theseconstraints when considering quantum states is quantumcloning, although it is simple to construct a quantumdevice that clones qubits which are promised to be inthe state | (cid:105) or | (cid:105) , it is not possible to design a universalquantum transformation that clones all qubit states [14].Another interesting example can be found in Ref. [15]where the authors consider universal not gates for qubits.Universal transformations on reversible quantum op-erations have been studied from several perspectives andmotivations such as gate discrimination [16, 17], cloningunitary operations [18], preventing quantum systems toevolve [19, 20], designing quantum circuits [3], learn- a r X i v : . [ qu a n t - ph ] A p r ing the action of a unitary [9, 21–24], transforming unit-ary operations into their complex conjugate [25], under-standing the role of causal order in quantum mechanics[10, 26], and others [27, 28]. Probabilistic exact transform-ations between multiple uses of reversible operations viaquantum circuits have been considered in Ref. [29] wherethe authors target transforming an arbitrary unitary op-eration into its inverse and in Ref. [30] where the authorsconsider the case where the unitary input-operation andthe unitary output-operation are two different repres-entations of the same group element. Also, Ref. [20, 31]consider the probabilistic exact circuits which act only inan auxiliary system which interacts to the target one viasome random Hamiltonian.This paper is focused on designing universal quantumcircuits which are not exclusively tailored for a partic-ular class of input-operations. That is, it should attainthe desired transformation for any d -dimensional unit-ary operation even if its description is not known. Inparticular, we focus on probabilistic heralded transform-ations between multiple uses of reversible operations. Inparticular, we focus on probabilistic heralded transform-ations between multiple uses of reversible operations.More precisely, we consider circuits which make useof a quantum measurement with an output associatedwith success and, when the success outcome is obtained,the transformation is implemented perfectly. We con-sider three classes of quantum protocols: parallel cir-cuits, where the input-operations can be performed sim-ultaneously, adaptive circuits where the input-operationsmay be used sequentially, and general protocols whichmay not be realisable by quantum circuits but are con-sistent with quantum theory when the use of the input-operations may be performed in an indefinite causal or-der [10, 11, 32]. We present a systematic approach basedon semidefinite programming that allows us to analysetransformations which is linear on quantum operations.We then analyse in details three particular transforma-tions, unitary transposition, unitary complex conjugation,and unitary inversion.Section II reviews results related to quantum circuitssuch as quantum supermaps, quantum combs, pro-cess matrices, and other important concepts. Section IIIpresents a general semidefinite programming (SDP) ap-proach to design optimal probabilistic exact quantum cir-cuits. Section IV introduces the concept of delayed input-states quantum circuits. Section V analyses circuits forimplementing unitary complex conjugation. Section VIanalyses circuits for unitary transposition. Section VIIanalyses circuits for unitary inversion and Sec. VIII con-cludes and discusses the main results. II. REVIEW ON HIGHER ORDER QUANTUMOPERATIONS AND SUPERMAPS
In this section we establish our notations and reviewhow to represent and analyse quantum circuits and trans- formations between quantum operations in terms of su-permaps. We refer to transformations between quantumstates as lower order operations ( i.e. , quantum chan-nels and quantum instruments) and transformationsbetween quantum operations ( e.g. , channels, instruments,quantum circuits ) as higher order operations, which willbe named as superchannels and superinstruements.
A. The Pills-Choi-Jamiołkowski Isomorphism
We start by reviewing the Choi isomorphism [33–35](also known as Pills-Choi-Jamiołkowski isomorphism),a useful way to represent linear maps and particularlyconvenient for completely positive ones. Let L ( H ) be theset of linear operators mapping a linear (Hilbert) space H to another space isomorphic to itself. This work onlyconsiders finite dimensional quantum systems, henceall linear spaces H are isomorphic to C d , d -dimensionalcomplex linear spaces. Any map (cid:101) Λ : L ( H in ) → L ( H out ) has a one to one representation via its Choi operatordefined by, C ( (cid:101) Λ ) : = ∑ ij (cid:101) Λ ( | i (cid:105)(cid:104) j | ) ⊗ | i (cid:105)(cid:104) j | ∈ L ( H out ⊗ H in ) , (1)where {| i (cid:105)} is an orthonormal basis. An important the-orem regarding the Choi representation is that a map (cid:101) Λ is completely positive (CP) if and only if its Choi oper-ator C ( (cid:101) Λ ) is positive semidefinite [35]. When the Choioperator C ( (cid:101) Λ ) of some map is given, one can obtain theaction of (cid:101) Λ on any operator ρ in ∈ L ( H in ) via the relation (cid:101) Λ ( ρ in ) = Tr in (cid:16) C ( (cid:101) Λ ) (cid:104) I out ⊗ ρ T in (cid:105)(cid:17) , (2)where ρ T in is the transposition of the operator ρ in in termsof the {| i (cid:105)} basis of H in We now present a useful mathematical identity re-garding the Choi isomorphism. Let U d , A and B be d -dimensional unitary operators . Any unitary quantumoperation (cid:102) U d ( ρ ) : = U d ρ U † d can be represented by itsChoi operator as C ( (cid:102) U d ) and a straightforward calcula-tion shows that (cid:104) A ⊗ B (cid:105) C ( (cid:102) U d ) (cid:104) A † ⊗ B † (cid:105) = C ( (cid:94) AU d B T ) . (3) B. Supermaps with single use of the input-operations
In quantum mechanics, physical states are representedby positive operators: ρ ∈ L ( H ) , ρ ≥
0, with unit trace: Symbols with a tilde represent linear maps. In principle, this identity also holds even when U d , A , and B are notunitary but general d -dimensional linear operators. Tr ( ρ ) =
1. In this language, universal transformationsbetween quantum states are represented by linear maps,to which we refer as just maps, (cid:101) Λ : L ( H in ) → L ( H out ) that are CP [1, 2]. Here, by universal we mean that themap (cid:101) Λ is defined for all quantum states ρ ∈ L ( H in ) andthe physical transformation can be applied to any of thesestates. Quantum channels are deterministic quantum op-erations and are represented by CP maps that preservethe trace of all quantum states ρ ∈ L ( H in ) . Probabilisticheralded universal transformations between quantumstates are represented by quantum instruments, a setof CP maps { (cid:102) Λ i } that sum to a trace preserving one, i.e. , (cid:101) Λ : = ∑ i (cid:102) Λ i is trace preserving (TP). Quantum in-struments describe measurements in quantum mechan-ics .When the set of instruments { (cid:102) Λ i } is performed, theoutcome i is obtained with probability Tr (cid:16)(cid:102) Λ i ( ρ ) (cid:17) , andthe state ρ is transformed to (cid:102) Λ i ( ρ ) Tr ( (cid:102) Λ i ( ρ ) ) .An important realisation theorem of quantum chan-nels is given by the Stinespring dilation [36] which statesthat every quantum channel (cid:101) Λ can be realised by firstapplying an isometric operation, i.e. , a unitary with aux-iliary systems and then discarding a part of the system.More precisely, every CPTP map (cid:101) Λ : L ( H in ) → L ( H out ) can be written as (cid:101) Λ ( ρ ) = Tr A (cid:48) (cid:16) U [ ρ ⊗ σ A ] U † (cid:17) (4)where σ A ∈ L ( A ) is some (constant) auxiliary state, U : H in ⊗ A → H out ⊗ A (cid:48) is a unitary acting on the mainand auxiliary system, and Tr A (cid:48) is a partial trace on somesubsystem A (cid:48) such that H out ⊗ A (cid:48) is isomorphic to H in ⊗A .Quantum instruments also have an important realisa-tion theorem that follows from Naimark’s dillation [1, 37].Every quantum instrument can be realised by a quantumchannel followed by a projective measurement, i.e. , ameasurement which all its POVM elements are project-ors, on some auxiliary system. More precisely, if { (cid:102) Λ i : L ( H in ) → L ( H out ) } represents an instrument, there ex-ist a quantum channel (cid:101) C : L ( H in ) → L ( H out ⊗ A ) and aprojective measurement given by { Π i } where Π i ∈ L ( A ) which satisfies: (cid:102) Λ i ( ρ ) = Tr A (cid:48) (cid:16) (cid:101) C ( ρ ) [ I H out ⊗ Π i ] (cid:17) . (5)We now define universal transformations betweenquantum operations in an analogous way in terms of Every instrument { (cid:102) Λ i } corresponds to a unique positive op-erator valued measure (POVM) measurement { M i } , M i ∈ L ( H in ) , M i ≥ ∑ i M i = I H in , such that Tr ( ρ M i ) = Tr (cid:16)(cid:102) Λ i ( ρ ) (cid:17) forevery state ρ ∈ L ( H in ) . The POVM { M i } can be written explicitly as M i = (cid:102) Λ † i ( I H out ) where (cid:102) Λ † i is the adjoint map of (cid:102) Λ i . linear supermaps [4]. Linear supermaps, to which wealso refer as just supermaps, are linear transformationsbetween maps. A supermap , (cid:102)(cid:102) S : [ L ( H ) → L ( H )] → [ L ( H ) → L ( H )] (6)represents transformations between input-maps (cid:103) Λ in : L ( H ) → L ( H ) to output ones (cid:103) Λ out : L ( H ) → L ( H ) .For instance, let (cid:102) U d ( ρ ) : = U d ρ U † d be the map associatedto the d -dimension unitary operation U d , the supermapthat transforms a unitary operation into its inverse isgiven by (cid:101)(cid:101) S ( (cid:102) U d ) = (cid:103) U − d .We say that a supermap (cid:101)(cid:101) S is TP preserving (TPP) if ittransforms TP maps into TP maps. Similarly, a supermapis CP preserving (CPP) when it transforms CP maps intoCP maps, and completely CP preserving (CCPP) if theevery trivial extension (cid:101)(cid:101) S ⊗ (cid:101)(cid:101) I , of (cid:101)(cid:101) S is CPP, where (cid:101)(cid:101) I ( (cid:101) Λ ) = (cid:101) Λ , ∀ (cid:101) Λ . A superchannel (cid:101)(cid:101) C is a supermap which respectstwo basic constraints: 1) it transforms valid quantumchannels into valid quantum channels (hence, CPP andTPP); 2) when performed to a part of a quantum channelthe global channel remains valid (hence, CCPP).Any one single use superchannel (cid:101)(cid:101) C has a determin-istic realisation in quantum theory and similarly to theStinespring dilation theorem, it can be shown that everysuperchannel admits a decomposition in terms of encoder and decoder of the form [4], (cid:101)(cid:101) C ( (cid:101) Λ ) = (cid:101) D ◦ (cid:104) (cid:101) Λ ⊗ (cid:102) I A (cid:105) ◦ (cid:101) E (7)where (cid:101) E : L ( H ) → L ( H ) ⊗ L ( H A ) is an isometrywhich maps an input-state ρ in ∈ L ( H ) to the spacewhere the map (cid:101) Λ acts and an auxiliary one L ( H A ) , (cid:102) I A isthe identity map on the auxiliary system ( i.e. (cid:102) I A ( σ A ) = σ A , ∀ σ A ∈ L ( H A ), (cid:101) D : L ( H ⊗ H A ) → L ( H ) is a unit-ary operation followed by a partial trace on a part of thesystem (see Fig. 2).The Choi representation allows us to describe any su-permap (cid:101)(cid:101) S : [ L ( H ) → L ( H )] → [ L ( H ) → L ( H )] asa map (cid:101) S : L ( H ⊗ H ) → L ( H ⊗ H ) acting on Choioperators. And by exploiting the Choi representationagain, we can represent any supermap (cid:101)(cid:101) S by a linear oper-ator S : = C ( (cid:101) S ) ∈ L ( H ⊗ H ⊗ H ⊗ H ) , which is use-ful to characterise the set of supermaps with quantumrealisations. In Ref. [3, 4] the authors show that a (cid:101)(cid:101) C isa superchannel if and only if its Choi representation C Symbols with a double tilde represent linear supermaps.
Figure 2. Every superchannel (cid:101)(cid:101) C : [ L ( H ) → L ( H )] → [ L ( H ) → L ( H )] transforming input-channels (cid:103) Λ in : L ( H ) → L ( H ) into output channels (cid:93) Λ out : L ( H ) → L ( H ) can be de-composed as (cid:101)(cid:101) C ( (cid:103) Λ in ) = (cid:101) D ◦ (cid:104)(cid:103) Λ in ⊗ (cid:102) I A (cid:105) ◦ (cid:101) E where the encoderoperation (cid:101) E : L ( H ) → L ( H ) ⊗ L ( H A ) is an isometry opera-tion, L ( H A ) is a space for some possible auxiliary system, andthe decoder (cid:101) D : L ( H ⊗ H A ) → L ( H ) is a unitary operationfollowed by a partial trace on a part of the system. respects C ≥ C = Tr C ⊗ I d ;Tr C = Tr C ⊗ I d ;Tr ( C ) = d d , (8)where d i is the dimension of the linear space H i . Weremark that although we introduce the general formalismwhere the dimensions d i may depend on i , we focus ourresults to the case where d i = d is independent of i .Supermaps with probabilistic a heralded quantum real-isation are given by superinstruments and play a sim-ilar role of instruments in higher order quantum opera-tions, that is, it formalises probabilistic transformationson quantum operations. Superinstruments are a set ofCCPP supermaps { (cid:101)(cid:101) C i } that sums to a superchannel. Theprobability of obtaining the outcome i when the super-instrument { (cid:101)(cid:101) C i } acts on the input-map (cid:101) Λ and input-state ρ is Tr (cid:16)(cid:104) (cid:101)(cid:101) C i (cid:16) (cid:101) Λ (cid:17)(cid:105) ( ρ ) (cid:17) and the state (cid:104)(cid:102)(cid:102) C i ( (cid:101) Λ ) (cid:105) ( ρ ) Tr (cid:16)(cid:104)(cid:102)(cid:102) C i ( (cid:101) Λ ) (cid:105) ( ρ ) (cid:17) is ob-tained. It follows from Ref. [38] that any superinstrumentcan be realised by a superchannel followed by a project-ive measurement, or equivalently, (cid:101)(cid:101) C i ( (cid:101) Λ ) = (cid:102) D i ◦ (cid:104) (cid:101) Λ ⊗ (cid:102) I A (cid:105) ◦ (cid:101) E , (9)here (cid:101) E : L ( H ) → L ( H ) ⊗ L ( H A ) is an isometry whichmaps an input-state ρ ∈ L ( H ) to the space where themap (cid:101) Λ acts and an auxiliary one L ( H A ) , (cid:102) I A is the identitymap on the auxiliary system ( i.e. , (cid:102) I A ( σ A ) = σ A , ∀ σ A ∈ L ( H A ) ), and the maps (cid:102) D i : L ( H ⊗ H A ) → L ( H ) forman instrument corresponding to a projective measure-ment. Figure 3. Illustration of a parallel (upper circuit) and an adapt-ive (lower circuit) protocol that transform a pair of quantumoperations (cid:102) Λ and (cid:102) Λ into an output one (cid:93) Λ out . C. Supermaps involving k input-operations In the previous section we have introduced supermapscorresponding to protocols involving a single use of aninput-operation. We now consider protocols transform-ing k , potentially different, operations into another. Let (cid:101)(cid:101) C be a superchannel which transforms k input-channels (cid:102) Λ j : L ( I j ) → L ( O j ) with j ∈ {
1, . . . , k } into an out-put one (cid:102) Λ : L ( I ) → L ( O ) . We also define the totalinput-state space as I : = (cid:78) kj = I j and the total outputspace state O : = (cid:78) kj = O j , hence (cid:101)(cid:101) C : [ L ( I ) → L ( O )] → [ L ( I ) → L ( O )] .Similarly to the single input-channel case, superchan-nels transforming k quantum operations are supermapswhich: 1) transform k valid quantum channels into avalid quantum channel; 2) when performed on a part of aquantum channel, the global channel remains valid. Dif-ferently from the k = k channels should be used in a definite causal order and itallows protocols which use the input-channels with anindefinite causal order [11].Protocols that can be implemented in the standardcausally ordered circuit formalism are referred to asquantum networks/quantum combs [3] or channels withmemory [39]. We divide these ordered circuits in twoclasses: a) parallel ones where k channels can be usedsimultaneously; b) adaptive ones where the k channelsare explored in a causal sequential circuit (see Fig. 3).Parallel protocols transforming k channels are verysimilar to single-channel superchannels presented in the We remark that here the subindex j stands for a label for the channel (cid:102) Λ j : L ( I j ) → L ( O j ) , not for some instrument element of an instru-ment { Λ j } . last subsection. Define (cid:101) Λ : = (cid:78) kj = (cid:102) Λ j , a superchannel (cid:101)(cid:101) C represents a parallel protocol if it can be written as (cid:101)(cid:101) C ( (cid:101) Λ ) = (cid:101) D ◦ (cid:104) (cid:101) Λ ⊗ (cid:102) I A (cid:105) ◦ (cid:101) E for some channels (cid:101) E : L ( I ) → L ( I ⊗ A ) and (cid:101) D : L ( O ⊗ A ) → O . It follows fromthe characterisation of Eq. (8) that a Choi operator C ∈ L ( I ⊗ (cid:78) kj = I j ⊗ (cid:78) kj = O j ⊗ O ) represents a parallelprotocol if and only if C ≥ O C = Tr OO C ⊗ I O d O ;Tr IOO C = Tr I IOO C ⊗ I I d I ;Tr ( C ) = d I d O . (10)Adaptive circuits can exploit a causal order relationbetween the channels (cid:102) Λ j to implement protocols thatcannot be done in a parallel way. A simple example isthe supermap that concatenates the channels (cid:102) Λ and (cid:102) Λ to obtain (cid:102) Λ ◦ (cid:102) Λ . This supermap has a trivial imple-mentation in an adaptive circuit (just concatenates thechannels) but cannot be implemented in a deterministic parallel scheme.A superchannel (cid:101)(cid:101) C : [ L ( I ) → L ( O )] → [ L ( I ) → L ( O )] corresponds to an adaptive circuit if it can be written as (cid:101)(cid:101) C ( (cid:101) Λ ) = (cid:101) D ◦ (cid:104) (cid:102) Λ k ⊗ (cid:102) I A (cid:105) ◦ (cid:102) E k ◦ . . . ◦ (cid:104) (cid:102) Λ ⊗ (cid:102) I A (cid:105) ◦ (cid:102) E (11)for some channels (cid:102) E : L ( I ) → L ( I ⊗ A ) , (cid:101) E i : L ( O i − ⊗ A ) → ( I i ⊗ A ) with i ∈ {
2, . . . , k } , and (cid:101) D : L ( O k ⊗ A ) → L ( O ) . A Choi operator C ∈ L ( I ⊗ (cid:78) kj = I j ⊗ (cid:78) kj = O j ⊗ O ) represents an adaptivesuperchannel if and only if [3, 39] C ≥ O C = Tr O k O C ⊗ I O k d O k ;Tr I i C ( i ) = Tr I i O i C ( i ) ⊗ I O i d O i , ∀ i ∈ { k , . . . , 2 } ;Tr I C ( ) = Tr I I C ( ) ⊗ I I d I Tr ( C ) = d I d O , (12)where C ( i ) : = Tr I i + O i + C ( i + ) for i ∈ {
1, . . . , k − } and C ( k ) : = Tr O k O C . Note that since we do not restrict the dimension of the auxiliarysystem A , all parallel protocols can be realised by an adaptive circuit. We now consider the most general protocols that trans-form k quantum channels into a single one. As men-tioned before, these superchannels may have an indef-inite causal order between the use of these k channels,hence they may not have an implementation in terms ofencoders and decoders in the standard quantum circuitformalism. Even without necessarily having a realisationby ordered circuits, it is possible to have a simple charac-terisation of these general superchannels. Before present-ing the necessary and sufficient condition for a general(possibly with an indefinite causal order) superchannels,it is convenient to introduce the trace and replace nota-tion introduced in Ref. [26]. Let A ∈ L ( H ⊗ H ) be ageneral linear operator, we define H A : = Tr H A ⊗ I H d H .A Choi operator C ∈ L ( I ⊗ I ⊗ I ⊗ O ⊗ O ⊗ O ) represents a general superchannel transforming k = C ≥ I O O C = I O O O C ; I O O C = O I O O C ; O C + O O O C = O O C + O O C ; IOO C = I IOO C ;Tr ( C ) = d I d O d O . (13)We remark that the bipartite process matrices presentedin Ref. [11, 26] correspond to a particular case of generalsuperchannels with two input-channels presented above.This correspondence is made by setting the dimensionof the linear spaces I and O as one. This occurs be-cause Ref. [11, 26] focus on superchannels that transformpairs of instruments into probabilities, not into quantumoperations. Also, the general superchannels presentedin Eq. (13) are equivalent to the general process matricespresented in Ref. [40] which uses the terminology com-mon past and common future to denote the spaces I and O , respectively.It is also possible to characterise general superchan-nels transforming k channels on terms of their Choi op-erators. For that, one can exploit the methods used inRef. [40] and [26] to characterise process matrices (seealso Ref. [41]) . Using such methods, we have charac-terised general superchannels which transforms k = C ∈ L ( I ⊗ I ⊗ I ⊗ I ⊗ O ⊗ O ⊗ O ⊗ O ) represents a general superchannel that transforms k = C ≥ I O I O O C = I O I O O O C ; I O I O O C = O I O I O O C ; I O I O O C = I O O I O O C ; I O O C + I O O O O C = I O O O C + I O O O C ; I O O C + O I O O O C = O I O O C + I O O O C ; I O O C + O O I O O C = O I O O C + O I O O C ; O C + O O O O C = O O C + O O C + O O C ++ O O O C + O O O C + O O O C ; IOO C = I IOO C ;Tr ( C ) = d I d O d O d O . (14)Similarly to the single use case, probabilistic heraldedprotocols are also represented by superinstruments. Su-perinstruments also admit a simple representation interms of their induced Choi operators. A set of paral-lel/adaptive /general superinstruments transforming k channels into another is given by a set of positive semi-definite operators C i ≥ C : = ∑ i C i is a validparallel/adaptive /general superchannel. The probab-ility of obtaining the outcome i when performing thesuperinstrument (cid:110) (cid:101)(cid:101) C i (cid:111) on k input-channels represen-ted by (cid:101) Λ : = (cid:78) kj = (cid:102) Λ j and the input-state ρ is given byTr (cid:16)(cid:104) (cid:101)(cid:101) C i (cid:16) (cid:101) Λ (cid:17)(cid:105) ( ρ ) (cid:17) . III. OPTIMAL UNIVERSAL QUANTUM CIRCUITS VIASDP
In this section we construct a systematic method todesign probabilistic heralded quantum circuits for trans-forming multiple uses of the same unitary operations.Let U d : L ( C d ) be a d -dimensional unitary operator and (cid:102) U d be a linear map representing the operation associ-ated to U d , that is, when the operation (cid:102) U d is appliedinto a quantum state ρ ∈ L ( C d ) the output is given by (cid:102) U d ( ρ ) = U d ρ U † d . We consider linear supermaps given by f : (cid:102) U d (cid:55)→ f ( (cid:102) U d ) mapping unitary operations into unitaryoperations. Our goal is to transform k uses of an arbit-rary (cid:102) U d into f ( (cid:102) U d ) with the highest heralded constantprobability p .From the results of the previous section, this transform-ation can be implemented via quantum circuits whenthere exists a superinstrument element i.e. , a CCPP lin-ear supermap, (cid:101)(cid:101) S such that (cid:101)(cid:101) S ( (cid:103) U ⊗ kd ) = p f ( (cid:102) U d ) for everyunitary operation (cid:102) U d (see Sec. II B). We stress that, eventhought we have presented an explict characterisationof superinstruments in Sec. II B, finding the optimal suc-cess probability for this transformation and its associated quantum circuit is, in general, a nontrival task. First,note that action of the supermap f is only described forunitary channels but the action of a superinstrumentelement (cid:101)(cid:101) S must be defined for any CP linear map. Thesupermap (cid:101)(cid:101) S can then be any CCPP linear supermap thatextends the action of f from unitary operations to gen-eral CP maps (see Ref. [42, 43] for a related lower-orderversion problem which consists of finding CP extentionsof linear maps defined on subspaces). Second, since k uses of the input-operation are available, it may be thecase that even if f does not have a linear CCPP extentionfor some number of uses k but it has for k > k (seeRef. [15, 44] for a lower-order analogue of this problemwhere multiple copies of the input- state can be used toimplement a linear positive non-CP map ).Before presenting our general approach we illustratethe subtleties of this extention problem by discussingthe universal channel complex conjugation studied inRef. [25]. Let (cid:101) Λ : H in → H out be a quantum channel withthe Kraus decomposition given by (cid:101) Λ ( ρ ) = ∑ i K i ρ K † i , wedefine the complex conjugate of (cid:101) Λ as the map which re-spects (cid:102) Λ ∗ ( ρ ) = ∑ i K ∗ i ρ K ∗ † i for every ρ where the complexconjugation of K i is made in a fixed orthonormal basis e.g. , the computational basis. One can show that, for anylinear spaces H in and H out with dimension greater thanor equal two, CCPP supermaps respecting (cid:101) Λ ⊗ k (cid:55)→ p (cid:102) Λ ∗ for all channels (cid:101) Λ necessarily have p = k ∈ N [45]. Hence it is not possible to design auniversal quantum circuit for probabilistic channel ad-joint. However, if one relaxes the requirements of generalchannels and seek for a quantum circuit that transformsonly unitary operations into their adjoints, universal com-plex conjugation can be implemented deterministicallyin a parallel circuit with makes k = d − k = d − f .We now present our SDP approach. Let (cid:110)(cid:101)(cid:101) S , (cid:101)(cid:101) F (cid:111) be asuperinstrument where the outcome of the element (cid:101)(cid:101) S indicates success and the outcome of (cid:101)(cid:101) F indicates failure.The problem of maximising the success probability oftransforming k uses of an arbitrary d -dimensional unitary Since we have imposed that f is linear, f is also implicitly definedfor linear combination of unitary operations. input-operation (cid:102) U d into f ( (cid:102) U d ) can be phrased as:max p s.t. (cid:101)(cid:101) S (cid:16) (cid:102) U d ⊗ k (cid:17) = p f ( (cid:102) U d ) , ∀ U d ; (cid:110)(cid:101)(cid:101) S , (cid:101)(cid:101) F (cid:111) is a valid superinstrument, (15)where the valid superinstrument representing a parallel,adaptive, or general protocols. Using the characterisa-tion presented in Sec. II, we can rewrite the above max-imisation problem only in terms of linear and positivesemidefinite constraints as: max p s.t. Tr IO (cid:32) S (cid:34) I I ⊗ C (cid:18) (cid:103) U ⊗ kd (cid:19) T IO ⊗ I O (cid:35)(cid:33) = p C ( f ( (cid:102) U d )) ∀ U d ; S , F ∈ L ( I ⊗ I ⊗ O ⊗ O ) , S ≥ F ≥ S + F is a valid superchannel. (16) Note that the maximisation problem presented in Eq. (16)must hold for all unitary operators U d and has infinitelymany constraints. This issue can be bypassed by notingthat due to linearity, it is enough to check these con-straints only for a set that spans the set spanned by Choioperators of unitary operations. That is, if we can write C ( (cid:103) U ⊗ kd ) = ∑ i α i C ( (cid:103) U ⊗ kd , i ) and it is true thatTr IO (cid:18) S (cid:20) I I ⊗ C (cid:16) (cid:103) U ⊗ kd , i (cid:17) T IO ⊗ I O (cid:21)(cid:19) = p C ( f ( (cid:103) U d , i )) ∀ i ;(17)we have thatTr IO (cid:18) S (cid:20) I I ⊗ C (cid:16) (cid:103) U ⊗ kd (cid:17) T IO ⊗ I O (cid:21)(cid:19) = ∑ i Tr IO (cid:18) S (cid:20) I I ⊗ α i C (cid:16) (cid:103) U ⊗ kd , i (cid:17) T IO ⊗ I O (cid:21)(cid:19) = p ∑ i α i C ( f ( (cid:103) U d , i ))= p C ( f ( (cid:102) U d )) . (18)Also, one can always find a finite set, in particular, a basis,of unitary operations { (cid:103) U d , i } that spans the set spannedby Choi operators of d -dimensional unitary operations, i.e. : span (cid:16) C ( (cid:103) U ⊗ kd ) | U d is unitary (cid:17) = span (cid:16) C ( (cid:103) U ⊗ kd , i ) | U d , i ∈ { U d , i } (cid:17) . (19)Explicitly obtaining a basis for the subspacespan (cid:16) C ( (cid:103) U ⊗ kd ) | U d is unitary (cid:17) is, in general, notstraightforward. For numerical purposes, this problemcan be tackled by sampling a large number of unitaries U d uniformly randomly (according to the Haar measure).If the dimension of this subspace is D , D unitariessampled uniformly will be linearly independent withunit probability. Since checking linear independence canbe done in an efficient way, we can construct a basis forthis set by sampling unitaries randomly until we cannotfind more linearly independent ones.Also note that the dimension D of the subspacespan (cid:16) C ( (cid:103) U ⊗ kd ) | U d is unitary (cid:17) may grow very fast with k and d , this will increase the number of constraints inthe SDP we have presented. Since having a large numberof constraints may make the SDP intractable for practicalpurposes (it may take a very long time to run the codeor to consume a very large amount of Random-accessmemory (RAM)), it is worth noticing that if one runsthe SDP (16) with a set of operators { U d , i } that do notform a basis for span (cid:16) C ( (cid:103) U ⊗ kd ) | U d is unitary (cid:17) , the solu-tion p of the SDP is not the maximal success probabilitybut an upper bound on the maximal success probabil-ity (it is the same SDP with fewer constraints). We alsopoint out that since the methods to solve an SDP alsoprovide the instrument element S that attains the max-imal success probability p , even if the set { U d , i } does notform a basis for span (cid:16) C ( (cid:103) U ⊗ kd ) | U d is unitary (cid:17) , it maystill be the case that the solution obtained is also theglobal optimal value . In order to check this hypothesiswe can extract the superinstruement element S of theSDP in which the operators { U d , i } that do not form abasis for span (cid:16) C ( (cid:103) U ⊗ kd ) | U d is unitary (cid:17) . Then, we gener-ate a basis { U (cid:48) d , j } for span (cid:16) C ( (cid:103) U ⊗ kd ) | U d is unitary (cid:17) andverify thatTr IO (cid:32) S (cid:34) I I ⊗ C (cid:18) (cid:93) U (cid:48) ⊗ kd , j (cid:19) T IO ⊗ I O (cid:35)(cid:33) = p C ( f ( (cid:103) U (cid:48) d , j )) (20)for every j .For the particular cases where the desired operationis unitary inversion, i.e. , f ( (cid:102) U d ) = (cid:103) U − d or the desired We thank Alastair Abbott for pointing this fact to us. When d = k =
2, we have applied this technique to tackle the unit-ary transposition and inversion problem. In this case, we have runour numerical SDPs only for a subset of the space of unitary channelsgenerated by span (cid:16) C ( (cid:103) U ⊗ ) | U is unitary (cid:17) . Numerically, we cansee that the linear space spanned by (cid:16) C ( (cid:103) U ⊗ ) | U is unitary (cid:17) has994 linearly independent unitary channels but we have only con-sidered a random subset containing 200 linear independent elementsof the form C ( (cid:103) U ⊗ ) in our calculations. After obtaining an upperbound to the problem, we have verified that the superinstrumentelement S transforms the full basis with 994 linearly independentunitary channels into their inverses, ensuring that the previous upperbound is tight. operation is unitary transposition , i.e. , f ( (cid:102) U d ) = (cid:102) U Td , theoptimal success probability p is always attainable by in-struments where the Choi operators S and F only havereal number components. That is, the operators S and F can be restricted to the field of real numbers with noloss of generality. To prove that, we first note that forevery unitary U d we have C (cid:16) (cid:103) U ⊗ kd (cid:17) ∗ = C (cid:18) (cid:93) U ∗ d ⊗ k (cid:19) where ∗ is complex conjugation in the computational basis. Wepresent the explicit proof for the unitary transpositioncase and the unitary inverse follows from the same steps.Assume that there exists a superinstrument with a suc-cess Choi operator S such thatTr IO (cid:18) S (cid:20) I I ⊗ C (cid:16) (cid:103) U ⊗ kd (cid:17) T IO ⊗ I O (cid:21)(cid:19) = p C ( (cid:102) U Td ) (21)holds for all unitaries U d . Direct calculation shows thatthe instrument defined by S ∗ attains the same perform-ance of S : Tr IO (cid:32) S ∗ (cid:34) I I ⊗ C (cid:18) (cid:103) U ⊗ kd (cid:19) T IO ⊗ I O (cid:35)(cid:33) (22) = Tr IO (cid:32) S ∗ (cid:34) I I ⊗ C (cid:18) (cid:103) U ⊗ kd (cid:19) T IO ⊗ I O (cid:35)(cid:33) ∗∗ (23) = Tr IO (cid:32) S ∗∗ (cid:34) I ∗I ⊗ C (cid:18) (cid:103) U ⊗ kd (cid:19) T IO∗ ⊗ I ∗O (cid:35)(cid:33) ∗ (24) = Tr IO (cid:32) S (cid:34) I I ⊗ C (cid:18) (cid:93) U ∗ d ⊗ k (cid:19) T IO ⊗ I O (cid:35)(cid:33) ∗ (25) = p C (cid:16) (cid:103) U ∗ d T (cid:17) ∗ (26) = p C (cid:16) (cid:102) U Td (cid:17) . (27) Since { S ∗ , F ∗ } represents a valid superinstrument, wecan construct the operators S (cid:48) : = S + S ∗ and F (cid:48) : = F + F ∗ which only have real number components.We have implemented our code using MATLAB [46]with the interpreter CVX [47] and tested with the solv-ers MOSEK, SeDuMi, and SDPT3 [48–50]. In Table I ofSec. VI F we apply this method to obtain the maximal suc-cess probability to transform k uses of a d -dimensionalunitary operation, i.e. , f ( (cid:102) U d ) = (cid:102) U Td under different con-straints. In Sec. VII C, we reproduce Table 1 of Ref. [29]which contains results for the maximal probability forunitary inversion i.e. , f ( (cid:102) U d ) = (cid:103) U − d . All our code areavailable at Ref. [51] and can be freely used, edited, anddistributed under the MIT license [52] and make extens-ive use of the toolbox QETLAB [53]. IV. DELAYED INPUT-STATE PROTOCOLS
In this section we define a particular subclass ofquantum circuits in which we refer to delayed input-state protocols. This class consists of circuits where the
Figure 4. Comparison between a standard quantum circuit (up-per circuit) and a delayed input-state protocol (lower circuit)that transforms general operations. In a delayed input-state pro-tocol, the input-state labelled by the space 1 is not used by theencoder operation (cid:101) E . The encoder only prepares a (potentiallyentangled) state which partially goes to the input-channel (cid:103) Λ in ,and then to the decoder channel (cid:101) D , which can perform a jointoperation between the input-state and the auxiliary system. input-state is provided after the input-operation whichwill be transformed (see Fig. 4). The concept of delayed-input-state generalises the class of supermaps consideredin the context of unitary learning and unitary store-and-retrieve problems [9, 21–24]. As we will show next, par-allel quantum circuits used for unitary transposition andunitary inversion can be assumed to be in the delayedinput-state form without loss of generality and the defin-ition of delayed input-state protocols is useful to provevarious theorems presented in this paper.Consider a scenario where Alice has k uses of a generalunitary operation (cid:102) U d until some time t . In a later time t , where (cid:102) U d cannot be accessed anymore, she would liketo implement f ( (cid:102) U d ) on some arbitrary quantum statechosen at time t . This scenario can be seen as a partic-ular case of the general unitary transformation problemwhere the input-state is only provided after the opera-tion (cid:102) U d . Let us start with the k = (cid:103) Λ in is allowed(see Fig. 4). In this single use case, every superchanneladmits a realisation in terms of a quantum circuit with anencoder and a decoder [4]. Let (cid:101)(cid:101) C be a superchannel trans-forming an input-operation (cid:103) Λ in : L ( H ) → L ( H ) into (cid:101)(cid:101) C ( (cid:103) Λ in ) = (cid:103) Λ out : L ( H ) → L ( H ) and ρ in ∈ L ( H ) bethe input-state on which she would like to apply (cid:103) Λ out . Aprotocol to implement the superchannel (cid:101)(cid:101) C can be realisedas following:1. Alice performs an encoder channel (cid:101) E : L ( H ) → L ( H ⊗ H A ) on the input-state ρ in ∈ L ( H ) .2. The input-operation (cid:103) Λ in : L ( H ) → L ( H ) is per-formed on a part of the state (cid:101) E ( ρ in ) ∈ L ( H ⊗ H A ) .3. The decoder (cid:101) D : L ( H ⊗ H A ) → L ( H ) is appliedto the state (cid:104)(cid:103) Λ in ⊗ (cid:102) I A (cid:105) (cid:16) (cid:101) E ( ρ in ) (cid:17) to obtain the finaloutput-state (cid:104)(cid:101)(cid:101) C ( (cid:103) Λ in ) (cid:105) ( ρ in ) = (cid:103) Λ out ( ρ in ) . (28)In a delayed input-state protocol, the encoder channel (cid:101) E does not have access to the input-state ρ in , since thisstate is only provided after the use of the operation (cid:103) Λ in .Instead of having an encoder channel, Alice must thenprepare a fixed state φ E ∈ L ( H ⊗ H A ) that is independ-ent of ρ in . More precisely, a superchannel (cid:102)(cid:102) C D representsa k = φ E ∈ L ( H ⊗ H A ) .2. The input-operation (cid:103) Λ in : L ( H ) → L ( H ) is per-formed on a part of the state φ E ∈ L ( H ⊗ H A ) prepared by Alice.3. The decoder (cid:103) D D : L ( H ⊗ H ⊗ H A ) → L ( H ) is applied to the state ρ in ⊗ (cid:104)(cid:104)(cid:103) Λ in ⊗ (cid:102) I A (cid:105) ( φ E ) (cid:105) toobtain the final output-state (cid:104) (cid:102)(cid:102) C D ( (cid:103) Λ in ) (cid:105) ( ρ in ) = (cid:103) Λ out ( ρ in ) . (29)We now consider parallel delayed input-state proto-cols with k > (cid:103) Λ in . By defin-ition, a parallel superchannel (cid:101)(cid:101) C : [ L ( I ) → L ( O )] → [ L ( I ) → L ( O )] that transforms k identical input-operations into another can be represented by an encoderchannel (cid:101) E : L ( I ) → L ( I ⊗ A ) and a decoder channel (cid:101) D : L ( O ⊗ A ) → L ( O ) such that (cid:101)(cid:101) C ( (cid:103) Λ ⊗ k ) = (cid:101) D ◦ (cid:104) (cid:103) Λ ⊗ k ⊗ (cid:101) I A (cid:105) ◦ (cid:101) E . (30)That is, in order to perform the output-operation (cid:103) Λ out = (cid:101)(cid:101) C ( (cid:103) Λ ⊗ k ) on an arbitrary input-state ρ in ∈ L ( I ) , we firstperform the encoder operation on ρ in , then the k uses of (cid:101) Λ on a part of the output of the encoder, and then thedecoder (cid:101) D : (cid:104)(cid:101)(cid:101) C (cid:16) (cid:103) Λ ⊗ k (cid:17)(cid:105) ( ρ in ) = (cid:101) D (cid:16)(cid:104) (cid:103) Λ ⊗ k ⊗ (cid:102) I A (cid:105) (cid:16) (cid:101) E ( ρ in ) (cid:17)(cid:17) . (31)In a delayed input-state protocol the encoder cannotnot make use of the input-state ρ in . Instead of an encoderchannel (cid:101) E we now consider some fixed (potentially en-tangled) quantum state φ E ∈ L ( I ⊗ A ) . On a delayedinput-state protocol, the decoder (cid:103) D D : L ( I ⊗ O ⊗ A ) → L ( O ) acts directly on input-state ρ in . We then say that a parallel superchannel (cid:102)(cid:102) C D represents a delayed input-state parallel protocol if can be written as (cid:104) (cid:102)(cid:102) C D (cid:16) (cid:103) Λ ⊗ k (cid:17)(cid:105) ( ρ in ) = (cid:103) D D (cid:16)(cid:104)(cid:104) (cid:103) Λ ⊗ k ⊗ (cid:102) I A (cid:105) ( φ E ) (cid:105) ⊗ ρ in (cid:17) ,(32)for some decoder channel (cid:103) D D : L ( I ⊗ O ⊗ A ) → L ( O ) and some state φ E ∈ L ( I ⊗ A ) . If we define a ψ (cid:103) Λ ⊗ k : = (cid:104) (cid:103) Λ ⊗ k ⊗ (cid:101) I A (cid:105) ( φ E ) , we can re-rewrite Eq. (32) as (cid:104) (cid:102)(cid:102) C D (cid:16) (cid:103) Λ ⊗ k (cid:17)(cid:105) ( ρ in ) = (cid:101) D (cid:16) ψ (cid:103) Λ ⊗ k ⊗ ρ (cid:17) . (33)Parallel delayed input-state superchannels (cid:102)(cid:102) C D alsohave a simple characterisation in terms its Choi oper-ator C D ∈ L ( I ⊗ I ⊗ O ⊗ O ) . Since the encoder actstrivially on the space L ( I o ) , it follows from the sametools used to characterise standard ordered circuits [3]that C D represents a parallel delayed input-state protocolif and only C D ≥ O C D = I I d I ⊗ Tr I OO C D ⊗ I O d O ;Tr ( C D ) = d I d O . (34)Or equivalently, C D respects the standard parallel super-map restrictions of Eq. (10) and alsoTr OO C D = Tr I OO C D ⊗ I I d I . (35)The formal definition and a simple Choi character-isation of adaptive delayed input-state protocols followstraightforwardly from the discussions of the parallelcase presented here . The case of superchannels withindefinite causal order is more subtle. Since they have noencoder/decoder ordered quantum circuit implementa-tion their physical interpretation is not evident. We letthe precise definition and the characterisation of non-causally ordered delayed input-state protocols for futureresearch.Probabilistic heralded parallel (adaptive) delayedinput-state protocols are given by superinstrumentswhose elements add to a superchannel representing aparallel (adaptive) delayed input-state protocol. It fol-lows from the circuit realisation of quantum instruments[38] that every parallel (adaptive) delayed input-stateprotocol can be realised by an encoder ( k − For adaptive protocols where the input-operation (cid:103) Λ in can be used k times one can also define the notion of k -delayed input-state pro-tocol, where the input-state is provided after the k th use of the input-operation (cid:103) Λ in . The characterisation of such protocols also followsfrom the discussion presented in this section and the methods presen-ted in Sec. II. (cid:101)(cid:101) S represents a super-instrument element of some higher order transformation,there exists a delayed input-state parallel superinstru-ment which, when successful, implements the action of (cid:101)(cid:101) S in a probabilistic heralded way. This theorem holdstrue even if the supermap (cid:101)(cid:101) S corresponds to an indefinitecausal order protocol. Intuitively, one can undersandthis theorem in terms of state teleportation and prob-abilistic heralded gate teleportation (see Sec. VI A for areview of gate teleportation). In order to “parallelise” anysuperinstrument one can use the gate teleportation to re-arrange the position of all input-operations in parallel.Also, one can always delay the use of the input state byexploiting the state teleportation protocol [54]. Althoughthe teleportation and gate teleportation protocol may fail,the success probability is strictly positive for any fixeddimension, ensuring that the success probability of theparallel circuit is non-zero. Lemma 1.
Let (cid:101)(cid:101) S : [ L ( (cid:78) ki = I i ) → L ( (cid:78) ko = O o )] → [ L ( I ) → L ( O )] be a supermap representing a general (pos-sibly with indefinite causal order) probabilistic protocol thatmakes k uses of a unitary operation (cid:102) U d and transforms tosome other unitary operation f ( (cid:102) U d ) with probability p U i.e., (cid:101)(cid:101) S (cid:16) (cid:102) U d ⊗ k (cid:17) = p U f ( (cid:102) U d ) . There exists a parallel delayed input-state protocol implementing the supermap (cid:101)(cid:101) S with a probabilitygreater than or equal to p U d total , where d total is the product of alllinear space dimensions, i.e., d total = ∏ ki = d I i ∏ ki = d O i Proof.
By assumption, (cid:101)(cid:101) S (cid:16) (cid:103) U ⊗ kd (cid:17) = p U f ( (cid:102) U d ) for all U d with probability p U . Since (cid:101)(cid:101) S must be a superinstrumentelement, the corresponding Choi operator S is positiveand respects Tr ( S ) ≤ d O d I , hence 0 ≤ d total S ≤ d I d O I ,where I ∈ L ( I ⊗ I ⊗ O ⊗ O ) is the identity oper-ator. Note that the Choi operator C P : = d I d O I rep-resents a valid parallel delayed input-state superchan-nel i.e. , it satisfies the parallel delayed input-state su-perchannel conditions of Eq. 34. We thus define thenew superinstrument via the Choi of its elements as S P : = d total S and F P : = d I d O I − d total S . It follows that F P ≥ S P + F P = C P = d I d O I is a valid paral-lel delayed input-state superchannel, hence the oper-ators S P and F P form a valid delayed-input state par-allel superinstrument. By linearity, we can verify that (cid:102)(cid:102) S P (cid:16) (cid:103) U ⊗ kd (cid:17) = d total (cid:101)(cid:101) S (cid:16) (cid:103) U ⊗ kd (cid:17) = p U d total f ( (cid:102) U d ) , ensuring thatwhen the output associated to S P is obtained, the probab- ilistic parallel delayed input-state protocol representedby the superinstrument elements S P and F P performsthe transformation of the supermap (cid:101)(cid:101) S with probability p U d total . V. UNIVERSAL UNITARY COMPLEX CONJUGATION
In this section we consider the problem of transform-ing k uses of an arbitrary d -dimensional unitary (cid:102) U d intoits complex conjugate (cid:102) U ∗ d for some fixed basis. We provethat when k < d − k = d − d -dimensional unitary operation (cid:102) U d into its complex conjugate (cid:102) U ∗ d . Hence, when com-bined with Ref. [25], our result reveals a characteristicthreshold property for exact unitary complex conjuga-tion: if k < d −
1, universal exact unitary complex conjug-ation is impossible (zero success probability), if k = d − Theorem 1 (Unitary complex conjugation: no-go) . Anyuniversal probabilistic heralded quantum protocol (includingprotocols without definite causal order) transforming k < d − uses of a d-dimensional unitary operation (cid:102) U d into itscomplex conjugate (cid:102) U ∗ d with probability p that does not dependon (cid:102) U d necessarily has p = , i.e., null success probability.Proof. From Lemma 1 we see that if there exists a super-instrument that transforms k uses of (cid:102) U d into its com-plex conjugate (cid:102) U ∗ d with some possibly smaller but stillpositive probability, there also exists a parallel super-instrument (cid:101)(cid:101) S that transforms k uses of (cid:102) U d into its com-plex conjugate (cid:102) U ∗ d with some positive probability p , i.e. , (cid:101)(cid:101) S ( (cid:103) U ⊗ kd ) = p (cid:102) U ∗ d . From the realisation theorem of the su-perinstruments (see Eq. (9) and Ref. [9]), there exist an iso-metry (cid:101) E : L ( I ) → L ( I ) ⊗ L ( A ) and an instrument ele-ment corresponding to success (cid:102) D S : L ( I ⊗ A ) → L ( O ) such that (cid:101)(cid:101) S ( (cid:103) U ⊗ kd ) = (cid:102) D S ◦ (cid:104) (cid:103) U ⊗ kd ⊗ (cid:102) I A (cid:105) ◦ (cid:101) E . (36)Let ρ IA By the Naimark dilation, the instrument element (cid:102) D S is given by (cid:102) D S ( ρ IA ) = Tr A (cid:16) D ρ IA D † (cid:17) (37) Reference [45] also proves that when d > k > (cid:102) U d are required for any non-null probabilisticheralded implementation. D ∈ L ( I ⊗ A ) . Set {| a (cid:105)} as a basis forthe auxiliary system A . The previous equation becomes (cid:102) D S ( ρ IA ) = ∑ a (cid:104) a | D ρ IA D † | a (cid:105) . (38)The operators D a : = (cid:104) a | D form a possible set of operatorsrealizing the instrument (cid:102) D S .Since we assume (cid:101)(cid:101) S (cid:16) (cid:103) U ⊗ kd (cid:17) = p (cid:102) U (cid:48) d , (cid:101)(cid:101) S (cid:16) (cid:103) U ⊗ kd (cid:17) must re-turn a pure state whenever the input-state is a pure state.The instrument element (cid:101) E is an isometry, hence its outputis always a pure state if the input-state is pure. This forces (cid:102) D S to preserve the purity of pure input-states, which inturn implies that (cid:104) a | D ρ IA D † | a (cid:105) must be the same for all a up to a proportionality constant. Let (cid:102) D a denote themap given by (cid:102) D a ( ρ IA ) : = D a ρ IA D † a = (cid:104) a | D ρ IA D † | a (cid:105) .The above argument shows that (cid:102) D a ◦ (cid:104) (cid:103) U ⊗ kd ⊗ (cid:102) I A (cid:105) ◦ (cid:101) E isalso a valid universal conjugation supermap.Without loss of generality we assume that k = d − not use any of the input-operations for the remaining cases of k < d −
2. Theimaginary unit √− | ψ (cid:105) ∈ I ∼ = C d and unitary operator U d ∈ L ( C ) must respect D a (cid:104) U ⊗ d − d ⊗ I (cid:105) E | ψ (cid:105) = e i φ ψ , Ud √ pU ∗ d | ψ (cid:105) , (39)where φ ψ , U d is a global phase that may depend on | ψ (cid:105) and U d . We see, however, that φ ψ , U d must be independent ofthe input-state. Set {| i (cid:105)} d − i = as the computational basisfor C d and the phase φ i , U d for when the input-state | ψ (cid:105) isequal to | i (cid:105) . Take a maximally entangled state | φ + d (cid:105) : = √ d ∑ d − j = | i (cid:105)| i (cid:105) in I ⊗ I R , where I R is a “copy” of I , i.e.,another d -dimensional quantum system left untouchedby (cid:101)(cid:101) S . We denote the corresponding phase by φ φ + d , U d .Let M U : = D a (cid:104) U ⊗ d − d ⊗ I (cid:105) E . Then, by linearity of M U and Eq. (39), we conclude that φ i , U d = φ φ + d , U d , hence nodependence on i . The subscript of φ ψ , U d for the input-state shall be omitted as φ U d We now parametrise the operators E and D a via theiraction on this basis as E | i (cid:105) I = ∑ (cid:126) i , i , a α (cid:126) i , i , a | i , . . . , i d − (cid:105) I ⊗ | a (cid:105) A ; O (cid:104) i | D a = ∑ (cid:126) i , i , a β (cid:126) i , i , a | i , . . . , i d − (cid:105) I ⊗ | a (cid:105) A , (40)where (cid:126) i = [ i , . . . , i d − ] is a vector such that i λ ∈{
0, . . . , d − } for any λ =
1, . . . , d −
2. Hereafter, werestrict to unitary operators U d that are diagonal in thecomputational basis such that U d = ∑ i e i θ i | i (cid:105)(cid:104) i | where θ i is any real number. For such diagonal U d its com-plex conjugate can be written as U ∗ d = ∑ i e − i θ i | i (cid:105)(cid:104) i | . ByEq. (39), (cid:104) i (cid:48) | D a (cid:104) U ⊗ d − d ⊗ I (cid:105) E | i (cid:48) (cid:105) = e i φ Ud √ p (cid:104) i (cid:48) | U ∗ d | i (cid:48) (cid:105) . (41)Substituting the definition (40), we obtain ∑ (cid:126) i , i , a α (cid:126) i , i , a β (cid:126) i , i , a e i (cid:104) ∑ d − λ = θ i λ (cid:105) = e i φ Ud √ pe − i θ i (cid:48) , (42)or, equivalently, ∑ (cid:126) i , i , a α (cid:126) i , i , a β (cid:126) i , i , a e i (cid:104) θ i (cid:48) + ∑ d − λ = θ i λ (cid:105) = e i φ Ud √ p , (43)for all i , i (cid:48) ∈ {
0, . . . , d − } and the diagonal U d . Notethat each U d corresponds to some choice of real numbers (cid:126) θ λ = [ θ , θ , . . . , θ d − ] and vice versa . Moreover, the left-hand side of Eq. (43) depends on i (cid:48) , but the right-had sidedoes not.In combinatorics, a weak composition of an integer n isa sequence of non-negative integers that sum to n . Theweak compositions that appear in this proof are that of d − d elements. The set of all such weak com-positions will be denoted by Γ and its elements ( i.e. , theindividual weak decomposition) by (cid:126) γ = [ γ , . . . , γ d − ] ,where the subscripts denote the elements of (cid:126) γ .In Eq. (43) the summation on i ranges between 0and d − (cid:126) i over all possible combinations of (cid:126) i =[ i , . . . , i d − ] where each i n ranges between 0 and d − ν l denote the number of times an integer l between 0and d − (cid:126) i and i . Recall that (cid:126) i consists of d − (cid:126) i and i in total are d − ∑ d − l = ν l = d −
1. Clearly, the sequence [ ν , . . . , ν d − ] belongs to Γ . With slight abuse of notation, let us set [ (cid:126) i , i ] = [ i , . . . , i d − , i ] . Each [ (cid:126) i , i ] corresponds a (cid:126) γ ∈ Γ .Each [ (cid:126) i , i ] with a given (cid:126) γ can be differentiated by an addi-tional parameter, say κ . More specifically, let K ( (cid:126) γ ) denotethe set of all sequences [ (cid:126) i , i ] with the weak decomposition (cid:126) γ . This extra parameter κ is then a natural number thatenumerates the sequences in K ( (cid:126) γ ) ( e.g. , via lexicographicordering). Thus the summation ∑ (cid:126) i , i , a in Eq. (43) can berelabelled as ∑ (cid:126) γ , κ , a . Introducing α (cid:48) (cid:126) γ : = ∑ κ , a α (cid:126) γ , κ , a β (cid:126) γ , κ , a ,we have ∑ (cid:126) γ α (cid:48) (cid:126) γ e i [ ∑ d − l = γ l θ l ] = e i φ Ud √ p . (44)Observe that for different (cid:126) γ , the functions e i [ ∑ d − l = γ l θ l ] arelinearly independent since θ i λ may take any value inthe reals. Each (cid:126) γ contains d elements and must sumup to d −
1. One of the elements, say γ l (cid:48) must be zerobecause the elements are non-negative. Set i (cid:48) = l (cid:48) inEq. (42) and use Eq. (44) to replace e i φ Ud √ p in the left-hand side of Eq. (42). Then all the terms that appear2in the right-hand side contain an exponent with a nonzero coefficient in front of θ l (cid:48) , while the coefficients of θ l (cid:48) are zero on the left-hand side. This equation can onlybe satisfied by setting α (cid:48) (cid:126) γ =
0, because exp ( i k θ ) andexp ( i k (cid:48) θ ) are linearly independent functions of θ , for anypair of distinct integers k and k (cid:48) . Thus, p = VI. UNIVERSAL UNITARY TRANSPOSITION
This section addresses the problem of universal unit-ary transposition. We consider probabilistic heraldedexact universal quantum protocols transforming k usesof a general d -dimensional unitary operation (cid:102) U d into itstranspose (cid:102) U d in terms of a fixed basis. When only paral-lel protocols are considered, we show that the maximalsuccess probability is exactly p s = − d − k + d − . Aslo, byexploiting ideas of the port-based teleportation [55], onecan design a delayed input-state parallel circuit that at-tains this maximal probability. When adaptive quantumcircuits are considered, we present an explicit protocolthat attains a success probability of p s = − (cid:16) − d (cid:17) (cid:100) kd (cid:101) ,which, for any constant dimension d , has an exponen-tial improvement over any parallel protocol. We thenanalyse quantum protocols with indefinite causal ordervia the SDP approach presented in Sec. III and show thatindefinite causal order protocols do have an advantageover causally ordered ones. A. Gate teleportation and single-use unitary transposition
Quantum teleportation is a universal protocol that canbe used to send an arbitrary d -dimensional quantumstate via classical communication assisted by quantumentanglement. We are going to describe the protocolfor pure states, as the extension to general mixed statesfollows from linearity. Suppose Alice holds the quditstate | ψ (cid:105) ∈ C d and shares with Bob a d -dimensionalmaximally entangled state | φ + d (cid:105) : = ∑ d − i = √ d | ii (cid:105) . In orderto “teleport” her state to Bob, Alice performs a generalBell measurement on | ψ (cid:105) and her share of the entangledstate and then sends the outcome of her measurementto Bob. The generalised Bell measurements have POVMelements given by M : = (cid:26)(cid:20)(cid:16) X id Z jd (cid:17) † ⊗ I d (cid:21) | φ + d (cid:105)(cid:104) φ + d | (cid:104)(cid:16) X di Z dj (cid:17) ⊗ I d (cid:105)(cid:27) i , j = d − i , j = ,(45) where X id : = d − ∑ l = | l ⊕ i (cid:105)(cid:104) l | ; Z jd : = d − ∑ l = ω jl | l (cid:105)(cid:104) l | , (46) Figure 5. Illustration of gate teleportation (upper circuit) andunitary transposition protocol (lower circuit). ω : = e π √− d , and l ⊕ i denotes l + i modulo d . The operat-ors X id and Z jd are known as the shift and clock operators,respectively, and can be seen as a generalisation of thequbit Pauli operators. Straightforward calculation showsthat, after Alice’s measurement, the state held by Bob isgiven by X id Z jd | ψ (cid:105) .After the measurement process is complete, Alicesends the measurement outcomes i and j of her jointmeasurement to Bob. Bob can then apply the unitaryoperation ( Z dj ) − ( X di ) − on his state to recover the state | ψ (cid:105) . Remark that, with probability p = d Alice obtainsthe outcomes i = j = U d on his half of themaximally entangled state before Alice performs the jointBell measurement, the final state is given by U d X id Z jd | ψ (cid:105) ,see Fig. 5. In this protocol, the operation U d performed byBob acts on the state | ψ (cid:105) held by Alice when the outcomesare i = j =
0, which happens with probability p = d . Gate teleportation can be represented as a quantumcircuit (see Fig. 5) and has applications in fault tolerantquantum computation [56].Our method to transform a single use of a general d -dimensional unitary operation (cid:102) U d into its transpose (cid:102) U Td is based on the circuit interpretation of gate teleporta-tion. The maximally entangled state respects the prop-erty I ⊗ A | φ + d (cid:105) = A T ⊗ I | φ + d (cid:105) for any linear operator A ∈ L ( C d ) . If Alice performs a general unitary U d onher half of the maximally entangled state, the state held The transposition is taken in the computational basis {| i (cid:105)} d − i = inwhich the maximally entangled state | φ + d (cid:105) = ∑ i √ d | ii (cid:105) is defined. U Td X id Z jd | ψ (cid:105) . With probab-ility p = d , the outcome i = j = U Td X id Z jd | ψ (cid:105) is equal to U Td | ψ (cid:105) , see Fig. 5. B. Port-based teleportation and parallel unitarytransposition
Port-based teleportation [55] has the same main goalas the standard state teleportation protocol. Alice wantsto “teleport” an arbitrary d -dimensional state | ψ (cid:105) to Bobwith classical communication assisted by shared entan-glement. The original motivation of Port-based teleport-ation is to perform a teleportation protocol that does notrequire a correction made via Pauli operators, but it canbe made simply by selecting some particular “port”. Forthat, it allows more general initial resource state andmore general joint measurements. The three main differ-ences of Port-based teleportation when compared to thestandard teleportation protocol presented in the previoussection can be summarised by:1. In port-based teleportation, instead of sharing a d -dimensional maximally entangled state, Alice andBob may share a general d k -dimensional entangledstates | φ (cid:105) ∈ (cid:16) C d ⊗ C d (cid:17) ⊗ k . This general entangledstate | φ (cid:105) can be seen as k pairs of qudits, referredto as “ports”.2. Instead of performing a generalised Bell measure-ment, Alice can perform a general joint measure-ment on | ψ (cid:105) and her half of the k entangled statesshared with Bob.3. Instead of performing the Pauli correction, Bobchooses a particular port based on Alice’s messageand discards the rest of the ports of his system.We note that since no Pauli correction is made, port-basedteleportation can only perform the teleportation task ap-proximately or probabilistically. In this paper we onlyconsider the probabilistic exact port-based teleportationwhere Alice performs a k + k outcomes are associated to the k ports she shareswith Bob and another outcome corresponding to failure.If Alice obtains the outcome of failure, she sends the fail-ure flag to Bob and the protocol is aborted. If she obtainsan outcome corresponding to some port l , she commu-nicates this corresponding outcome to Bob and the state | ψ (cid:105) is teleported to Bob’s port labelled by l .The optimal probabilistic single port ( k =
1) case isessentially the standard state teleportation. Considerthe case where Alice and Bob share the d -dimensionalmaximally entangled state | φ + d (cid:105) . If we set the meas-urement performed by Alice as M = | φ d (cid:105)(cid:104) φ d | and M fail = I − | φ d (cid:105)(cid:104) φ d | , with probability p = d , the state | ψ (cid:105) is obtained in the single port 1, and with probability p F = − d the protocol fails. Reference [57] shows that the optimal probabilisticport-based gate teleportation protocol for any dimen-sion d and number of states k with success probability p = − d − k + d − . Reference [57] also characterises theoptimal d k -dimensional shared entangled state and theoptimal joint measurement Alice must perform. The op-timal state resource state is described by exploiting theSchur-Weyl duality C d ⊗ k ∼ = (cid:77) µ ∈ irrep ( U ⊗ k ) C dim ( µ ) µ ⊗ C m µ , (47)where irrep ( U ⊗ k ) is the set of all irreducible representa-tions µ of the group of special unitary SU ( U d ) containedin the decomposition U ⊗ k and m µ is the multiplicity ofthe representation µ . The optimal resource state used forport-based teleportation can be written as | φ PBT (cid:105) : = (cid:77) µ ∈ irrep ( U ⊗ k ) √ p µ | φ + ( µ ) (cid:105) ⊗ | ψ m µ (cid:105) , (48)where | φ + ( µ ) (cid:105) : = (cid:112) dim ( µ ) ∑ i | i µ i µ (cid:105) ∈ C dim ( µ ) µ ⊗ C dim ( µ ) µ (49)is the maximally entangled state on the linear space ofthe irreducible representation µ , { p µ } is a probability dis-tribution, and | ψ m µ (cid:105) ∈ C m ( µ ) ⊗ C m ( µ ) is a pure quantumstate.In Sec. VI A we have exploited the standard state gateteleportation to construct a protocol that can be used totransform a general unitary U d into its transpose U Td . Wenow exploit port-based gate teleportation to construct aparallel protocol that transforms k uses of U d to obtainits transpose.The first important observation is that the state | φ PBT (cid:105) (Eq. (48)) respects U ⊗ kd ⊗ I | φ PBT (cid:105) = I ⊗ U T ⊗ k d | φ PBT (cid:105) . (50)This identity holds true because every tensor product of k unitaries U d can be decomposed as U ⊗ kd ∼ = (cid:77) µ ∈ irrep ( U ⊗ k ) U ( µ ) ⊗ I m µ (51)for some unitaries U ( µ ) acting on the irreducible rep-resentation space C dim ( j ) j [57]. Hence, similarly to thecase of the single use unitary transposition, we can adaptport-based gate teleportation to obtain a general protocol Here the symbol ∼ = is used to ephasise that the Eq. (51) is true up toan isometry. Figure 6. Illustration of the modified port-based teleportationprotocol that makes k = d -dimensionalunitary operation (cid:102) U d where the state | ψ PBT (cid:105) is described inEq. (48) and the decoder (cid:101) D simply selects a particular portaccordingly to the outcome of the joint measurement M . Theupper circuit exploits port-based gate teleportation to store k = (cid:102) U d and returns a single use ofit with probability p . The lower circuit exploits port-based gateteleportation to transform k uses of (cid:102) U d into a single use of itstranspose (cid:102) U Td . The upper and lower circuits are successful withprobability p = − d − k + d − . to transform k uses of a general unitary operation (cid:102) U d into its transpose (cid:102) U Td . It is enough to perform the opera-tion (cid:102) U d on each of her half of entangled qudit states (seeFig. 6). We will show in Sec. VI D that this protocol is alsooptimal in terms of success probability. C. Review on probabilistic exact unitary learning
We make a brief summary of problem known as unit-ary learning (also known as storage and retrieval ofunitary operations) [9, 21–24]. As we will show inSec. VI D, the problem of probabilistic unitary learningis closely connected to the problem of parallel unitarytransposition and results related to unitary learning willbe useful to prove the optimality of our parallel unit-ary transposition protocol. Suppose that, until sometime t , Alice has access to k uses of some general d -dimensional unitary operation U d of which the descrip-tion is not provided. In a later moment t , where Alicecannot access U d any more, she wants to implement theaction of this unitary on some general quantum state ρ chosen at time t . A parallel strategy to succeedin this task is to perform the k uses of U d on parts ofan entangled quantum state φ E before t to obtain aquantum state ψ M : = (cid:104) U ⊗ kd ⊗ I (cid:105) φ E (cid:104) U † ⊗ kd ⊗ I (cid:105) . Alicethen saves this state ψ M until a later time t where sheperforms a global decoder operation (cid:101) D on the state ψ M together with the target state ρ , which is desired to sat-isfy (cid:101) D ( ψ M ⊗ ρ ) = U d ρ U † d . References [9, 21–23] con-sider deterministic non-exact unitary learning protocolsand analyse strategies that simulate the action of (cid:102) U d withthe maximal average fidelity, while Ref. [24] considersprobabilistic heralded protocols that can be used to re-trieve (a single use of) (cid:102) U d exactly but may fail with someprobability.The unitary learning problem described above can berephrased as the problem of finding delayed input-stateprotocols that transform k uses of a general unitary oper-ation (cid:102) U d into itself. In Sec. VI D we present a one-to-oneconnection between probabilistic unitary learning pro-tocols and delayed input-state parallel protocols trans-forming k uses of a general unitary operation (cid:102) U d into itstranspose (cid:102) U Td . Essentially, we show that any probabil-istic unitary learning with success probability p can betranslated into a parallel unitary transposition protocolwith success probability p . This one-to-one connection isrelated to the fact that the optimal resource state used forunitary learning and the optimal resource state used forparallel delayed input-state unitary transposition can beboth chosen as a state | φ (cid:105) which respects the property U ⊗ kd ⊗ I | φ (cid:105) = I ⊗ U T ⊗ k d | φ (cid:105) , (52)as shown in next subsection. D. Optimal parallel unitary transposition protocols
We show how any parallel protocol that can be used totransform k copies of a general unitary operation (cid:102) U d intoits transpose (cid:102) U Td can be adapted into a delayed input-state protocol keeping the same success probability. In principle, one may also consider adaptive protocols to performbetter in the unitary learning problem. In an adaptive protocol, onecan perform different enconder operations in between the use of theunitary to create more general protocols. One may also considerprotocols where the unitaries U d are used without a definite causalorder. References [9, 24] show that, for the unitary learning problem,the protocol with highest success probability (exact implementation)and highest expected fidelity (deterministic implementation) canalways be parallelised. We note that although the main goal is to obtain a decoder chan-nel (cid:101) D and entangled state φ E such that (cid:101) D ( ψ M ⊗ ρ ) = U d ρ U † d where ψ M : = (cid:104) U ⊗ kd ⊗ I (cid:105) φ E (cid:104) U † ⊗ kd ⊗ I (cid:105) , the unitary learning task cannot berealised in a deterministic and exact way for a general unitary U d . Lemma 2.
Any parallel probabilistic heralded protocol trans-forming k copies of a general unitary (cid:102) U d into (cid:102) U Td with a con-stant probability p can be converted to a delayed input-stateparallel protocol with the same probability p.Proof. Let S be the Choi operator of the superinstrumentelement associated to success and F be the Choi operatorof the superinstrument element associated to failure. Su-perinstrument element S transforms k copies of (cid:102) U d into (cid:102) U Td with probability p , i.e. ,Tr IO (cid:18) S (cid:20) I I ⊗ C (cid:16) (cid:103) U ⊗ kd (cid:17) T ⊗ I O (cid:21)(cid:19) = p C ( (cid:102) U Td ) ∀ U d ,(53)and S + F is a valid parallel superchannel.Since S transforms every unitary operator into its trans-pose, we can make the change of variable U d (cid:55)→ BU d A T where A and B are arbitrary d -dimensional unitary op-erators. With that, unitary transposition can be seen as (cid:0) BU d A T (cid:1) ⊗ k (cid:55)→ p ( BU d A T ) T = pAU Td B T . Our goal nowis to show that if S respects Eq. (53), any operator S (cid:48) re-specting S (cid:48) = (cid:104) A I ⊗ B ∗⊗ k I ⊗ A ∗⊗ k O ⊗ B O (cid:105) S (cid:104) A † I ⊗ B T ⊗ k I ⊗ A T ⊗ k O ⊗ B † O (cid:105) (54)satisfiesTr IO (cid:18) S (cid:48) (cid:20) I I ⊗ C (cid:16) (cid:103) U ⊗ kd (cid:17) T ⊗ I O (cid:21)(cid:19) = p C ( (cid:102) U Td ) ∀ U d .(55)To prove this fact, first note that theidentity presented in Eq. (3) implies that C ( (cid:94) AU Td B T ) = [ A ⊗ B ] C ( (cid:102) U Td ) (cid:2) A † ⊗ B † (cid:3) , and C (cid:32)(cid:20) (cid:94) BU d A T (cid:21) ⊗ k (cid:33) = (cid:104) B ⊗ k ⊗ A ⊗ k (cid:105) C ( (cid:103) U ⊗ kd ) (cid:104) B † ⊗ k ⊗ A † ⊗ k (cid:105) ,(56) which implies C (cid:32)(cid:20) (cid:94) BU d A T (cid:21) ⊗ k (cid:33) T = (cid:104) B ∗ ⊗ k ⊗ A ∗ ⊗ k (cid:105) C ( (cid:103) U ⊗ kd ) T (cid:104) B T ⊗ k ⊗ A T ⊗ k (cid:105) .(57) Substituting Eq. (57) and C ( (cid:94) AU Td B T ) = [ A ⊗ B ] C ( (cid:102) U Td ) (cid:2) A † ⊗ B † (cid:3) in Eq. (53)we obtain Tr IO (cid:16) S (cid:104) I I ⊗ (cid:104) B ∗ ⊗ k ⊗ A ∗ ⊗ k (cid:105) C ( (cid:103) U ⊗ kd ) T (cid:104) B T ⊗ k ⊗ A T ⊗ k (cid:105) ⊗ I O (cid:105)(cid:17) = (cid:2) A I ⊗ B O (cid:3) C ( (cid:102) U Td ) (cid:104) A † I ⊗ B † O (cid:105) . (58) If we apply the operator A † I ⊗ B † O on the left side andthe operator A I ⊗ B O on the right side of Eq.(58) anduse the cyclic property of the trace, we find that S can besubstituted by S (cid:48)(cid:48) : = (cid:104) A † I ⊗ B T ⊗ k I ⊗ A T ⊗ k O ⊗ B † O (cid:105) S (cid:104) A I ⊗ B ∗⊗ k I ⊗ A ∗⊗ k O ⊗ B O (cid:105) . (59) Since A and B are arbitrary unitary operators, we cantake the invertible transformations A † (cid:55)→ A and B † (cid:55)→ B to obtain the symmetry of Eq. (54).The symmetry presented in Eq. (54) motivates thedefinition of a Haar measure “twirled” map (cid:101) τ ( S ) : = (cid:90) Haar (cid:104) A I ⊗ B ∗⊗ k I ⊗ A ∗⊗ k O ⊗ B O (cid:105) S (cid:104) A † I ⊗ B T ⊗ k I ⊗ A T ⊗ k O ⊗ B † O (cid:105) d A d B . (60)We now define a twirled version of the superinstrumentas S τ : = (cid:101) τ ( S ) and F τ : = (cid:101) τ ( F ) , which respects the condi-tions of valid superinstruments and S τ also transforms k uses of any (cid:102) U d into (cid:102) U Td with probability p . We nownotice that both S τ and F τ respectsTr OO S τ = (cid:90) Haar (cid:104) A I ⊗ B ∗⊗ k I (cid:105) Tr OO ( S ) (cid:104) A † I ⊗ B T ⊗ k I (cid:105) d A d B ∝ I I ⊗ Tr I OO S τ (61)since the identity is the only operator that commuteswith all unitary operations (Schur’s lemma). It followsthen that the superchannel C τ : = S τ + F τ respects theconditions of a parallel delayed input-state protocol. Lemma 3.
For every delayed input-state parallel protocoltransforming k uses of a general unitary operation (cid:102) U d into itstranspose (cid:102) U Td with success probability p that is independent of (cid:102) U d , there exists a probabilistic unitary learning protocol witha success probability p.Conversely, for every probabilistic unitary learning protocolwith a success with probability p that is independent of (cid:102) U d ,there exists a delayed input-state parallel protocol transformingk uses of a general unitary operation (cid:102) U d into its transpose (cid:102) U Td with a constant success probability p.Proof. We start by showing how one can adapt a parallelprotocol transforming k uses of a general unitary opera-tion (cid:102) U d into its transpose (cid:102) U Td into a unitary learning onewith the same success probability.Let S be the Choi operator of the superinstrument ele-ment associated to success and F be the Choi operator ofthe superinstrument element associated to failure. Super-instrument element S transforms k copies of (cid:102) U d into (cid:102) U Td with probability p , i.e. ,Tr IO (cid:18) S (cid:20) I I ⊗ C (cid:16) (cid:103) U ⊗ kd (cid:17) T ⊗ I O (cid:21)(cid:19) = p C ( (cid:102) U Td ) ∀ U d ,(62)Lemma 2 states that this protocol can be converted tohave a delayed input-state and without lost of generality,the superchannel C = S + F respects the commutationrelation (cid:104) C , A ∗I ⊗ B ⊗ k I ⊗ A ⊗ k O ⊗ B ∗O (cid:105) = A , B ∈ SU ( d ) .When a Choi operator C represents a delayed input-state protocol, the operator C I : = Tr I OO C is propor-tional to the reduced state Tr A ( φ E ) of the state φ E ∈ L ( I ⊗ A ) prepared by Alice before the use of the input-operations . From the commutation relation in Eq. (63),we see that C I respects (cid:104) C I , B ⊗ k I (cid:105) =
0. (64)The Schur-Weyl duality states that k identical d -dimensional unitaries B can be decomposed as (seeSec. VI B) B ⊗ k ∼ = (cid:77) µ ∈ irrep ( U ⊗ kd ) B ( µ ) ⊗ I m ( µ ) , (65)where B ( µ ) ∈ L (cid:16) C dim ( µ ) µ (cid:17) is a unitary operator, and I m ( µ ) is the identity on the multiplicity space C m ( µ ) .Since the reduced state Tr A ( φ E ) respects the relation (cid:104) Tr A ( φ E ) , B ⊗ k (cid:105) =
0, Schur’s lemma ensures that thereduced encoder state has the form ofTr A ( φ E ) ∝ (cid:77) µ I µ ⊗ ρ m µ , (66)where I µ is the identity on the the linear space C dim ( µ ) µ and ρ m µ is some state on the multiplicity space of µ .Without loss of generality, we can assume that φ E = | φ E (cid:105)(cid:104) φ E | is a pure state with a reduced state that respectsEq. (66). It follows then that | φ E (cid:105) can be written as | φ E (cid:105) : = (cid:77) µ ∈ irrep ( U ⊗ k ) √ p µ | φ + ( µ ) (cid:105) ⊗ | ψ m µ (cid:105) , (67)where | φ + ( µ ) (cid:105) : = (cid:112) dim ( µ ) ∑ i | i µ i µ (cid:105) ∈ C dim ( µ ) µ ⊗ C dim ( µ ) µ (68)is the maximally entangled state on the linear space ofthe irreducible representation µ , { p µ } is a probabilitydistribution, and | ψ m µ (cid:105) ∈ C m ( µ ) ⊗ C m ( µ ) are some puri-fications of ρ m µ .We now make an important observation. Although thestate | φ E (cid:105) is not the maximally entangled state, it respects U ⊗ kd ⊗ I | φ E (cid:105) = I ⊗ U T ⊗ k d | φ E (cid:105) . (69) See Fig. 4 for a pictorial illustration for the case k =
1. Let φ E ∈ L ( H ⊗ H A ) be the state created by the encoder of the delayed input-state protocol of Fig. 4. In this case, C : = Tr C is proportional thereduced state Tr A φ E . This identity holds true because any tensor product of k identical unitaries U d can be decomposed as U ⊗ kd ∼ = (cid:77) µ ∈ irrep ( U ⊗ k ) U ( µ ) ⊗ I m µ (70)for some unitaries U ( µ ) acting on the invariant repres-entation space C dim ( j ) j . Any delayed input-state protocolthat can be used for unitary transposition can be usedfor unitary learning, since it is enough to perform theunitaries U ⊗ kd on the “other” half of the entangled state | φ E (cid:105) on which the joint operation is not performed.We now show how to transform probabilistic unitarylearning protocols to heralded unitary transposition pro-tocols. In Ref. [24], the authors have shown that, withoutloss of generality, any probabilistic unitary learning pro-tocol can be made parallel and, moreover, with the en-tangled state | φ E (cid:105) ∈ L ( I ⊗ A ) which respects the prop-erty U ⊗ kd ⊗ I | φ E (cid:105) = I ⊗ U T ⊗ k d | φ E (cid:105) . (71)Hence, if we perform the unitary operations U ⊗ kd intothe half of the entangled state | ψ (cid:105) on which the jointmeasurement is performed, the unitary recovered afterthe learning protocol will be U Td instead of U d .We are now in position to prove that the protocol basedon port-based gate teleportation presented in Sec. VI B isoptimal. Theorem 2 (Optimal parallel unitary transposition) . Themodified port-based gate teleportation protocol can be usedto transform k uses of an arbitrary d-dimensional unitaryoperation (cid:102) U d into its transpose (cid:102) U Td with success probabilityp = − d − k + d − in a parallel delayed input-state protocol.Moreover, this protocol attains the optimal success probabilityamong all parallel protocols with probability p that does notdepend on (cid:102) U d .Proof. As shown above, the identity U ⊗ k ⊗ I | φ PBT (cid:105) = I ⊗ U T ⊗ k | φ PBT (cid:105) ensures that port-based gate teleporta-tion can be used to construct a delayed input-state par-allel protocol that obtains U Td with k uses of U d withprobability p = − d − k + d − .Lemma 3 shows that any protocol transforming k usesof (cid:102) U d into its transpose (cid:102) U Td in a parallel protocol withprobability p can be used to succesfully “learn” theinput-operation (cid:102) U d with probability p and k uses. Ref-erence [24] shows that the optimal protocol for unitarylearning a unitary U d with k uses cannot have constantprobability greater than p = − d − k + d − , which boundsour maximal probability of success and finishes the proof.7 Figure 7. A flowchart illustrating the adaptive unitary trans-pose protocol.
E. Adaptive unitary transposition protocols
In this subsection we present an adaptive circuit thattransforms k uses of an arbitrary d -dimensional unitaryoperation (cid:102) U d into a single use of its transpose (cid:102) U Td withheralded probability p = − (cid:16) − d (cid:17) (cid:100) kd (cid:101) (see Fig. 7).1. We start by making a single use of the input-operation (cid:102) U d to implement the probabilistic her-alded transposition protocol based on gate teleport-ation described in Sec. VI D. When the generalisedBell measurement returns the outcomes i and j ,the operator U d is transformed into V = U Td X id Z jd ,where X id and Z jd are the clock and shift operators,respectively (see Eq. (46)).2. If both outcomes i and j correspond to the identityoperator, i.e. , i = j =
0, we have V = U Td andwe stop the protocol with success. If some otheroutcome is obtained, we make d − U d toimplement the unitary complex conjugate protocol[25] to obtain U ∗ d . We then apply X id − Z jd − U ∗ d into V to “cancel” the transformation of step 1 to obtainidentity operator (cid:20) X id − Z jd − U ∗ d U Td Z jd X id = I d (cid:21) .3. Go to step 1.We see that step 1 fails returning (cid:102) U Td with probabil-ity (cid:16) − d (cid:17) and we need in total d uses of the input-operation (cid:102) U d to complete steps 1 and 2. These steps maybe repeated up to (cid:100) kd (cid:101) times, hence they lead to a successprobability of p = − (cid:16) − d (cid:17) (cid:100) kd (cid:101) . d = k = = = = k = = ≈ ≈ k = = ≈ d = k = ≈ ≈ ≈ k = = ≈ ≈ Table I. Table with optimal success probability we have ob-tained for heralded protocols transforming k uses of (cid:102) U d into asingle use of its transpose (cid:102) U Td . The values in blue were provedanalytically and the values in black were obtained via numerialSDP optimisation. F. Optimal protocols via SDP formulation and indefinitecausal order advantage
We apply the SDP methods obtained in Sec. III to thecase of unitary transposition and present the optimalsuccess probability in Table I. By checking Table I oneobserves that the adaptive circuit we have presented inSec. VI E is not optimal. One possible intuitive under-standing is that the adaptive protocol we have presentedin Sec. VI E “wastes” d − (cid:102) U d to recover the input-state. We also notice that indefinitecausal order protocols provide a strictly large successprobability when compared to causally ordered ones. Itis interesting to observe that although indefinite causalorder protocols have been reported useful in tasks suchas non-signalling channel discrimination [58], quantumcomputation [32], and quantum channel capacity activ-ation [27, 28], this is the first time that indefinite causalorder protocols outperform causally ordered ones whenmultiple uses of the same unitary input-operation aremade. In those previous examples cited, the advantageof indefinite causal order was obtained by exploitingthe quantum switch [10], a process which is not usefulin our task of unitary channel transformation, since thequantum switch would transform k uses of the any unit-ary operation (cid:102) U d into simple k concatenations of (cid:102) U d , orequivalently, a single use of (cid:102) U kd . Our results for indefinitecausal order then reveals the existence of a different classof indefinite causal order protocols, similarly to the onereported for unitary inverse in Ref. [29]. VII. UNIVERSAL UNITARY INVERSION PROTOCOLS
We now address the problem of transforming k usesof a general d -dimensional unitary operation (cid:102) U d into asingle use of its inverse (cid:103) U − d with probabilistic heraldedquantum circuits. We have presented our adaptive circuitin Ref. [29] and here we present a parallel implementa-tion and provide more details on the adaptive circuit.8Before presenting our protocols we prove that, simil-arly to the complex conjugation case, any protocol per-forming exact universal unitary inversion with k < d − (cid:102) U d necessarily hasnull success probability. Also, this no-go result also holdseven when protocols with indefinite causal order are con-sidered. Theorem 3 (Unitary inversion: no-go) . Any universalprobabilistic heralded quantum protocol (including protocolswithout definite causal order) transforming k < d − usesof a d-dimension unitary operation (cid:102) U d into a single use of itsinverse (cid:103) U − d with success probability p that does not dependon U d necessarily has p = i.e., null success probability.Proof. Assume that there exists a quantum protocol trans-forming k uses of a general d -dimensional unitary op-eration (cid:102) U d into its inverse (cid:103) U − d with a non-zero successprobability p . We can then exploit the single-use unitarytransposition protocol presented in Sec. VI A to obtain (cid:102) U ∗ d with success probability p / d >
0, which contradictsLemma 1.
A. Parallel unitary inversion protocols
We start by showing that, similarly to universal paral-lel transposition, any universal parallel unitary inversionprotocol can be made in a delayed input-state way.
Lemma 4.
Any parallel probabilistic heralded parallel pro-tocol k uses of a general unitary (cid:102) U d into a single use of itsinverse (cid:103) U − d with constant probability p can be conversed to adelayed input-state parallel protocol with the same probabilityp.Proof. The proof follows the same steps as the one inTheorem 2. The only difference is that for unitary trans-position, the superinstrument element S can be chosenas an operator that commutes with unitaries of the form A I ⊗ B ∗⊗ k I ⊗ A ∗⊗ k O ⊗ B O and for unitary inversion S can be chosen as an operator which commutes with allunitaries of the form of A I ⊗ B ⊗ k I ⊗ A ⊗ k O ⊗ B O .We are now in conditions to present a universal cir-cuit for parallel unitary inversion and also to obtain anupper bound on the maximal success probability. Ourprotocol makes use of the unitary complex conjugationand unitary transposition and it is proven to be optimalfor qubits. Theorem 4 (Universal unitary inverse) . There exists a par-allel delayed input-state probabilistic quantum circuit thattransforms k uses of an arbitrary d-dimensional unitary opera-tion (cid:102) U d into a single use of its inverse (cid:103) U − d with success prob-ability p S = − d − k (cid:48) + d − where k (cid:48) : = (cid:98) kd − (cid:99) is the greatestinteger that is less than or equal to kd − . The maximal success probability transforming k uses of anarbitrary d-dimensional unitary operation (cid:102) U d into a singleuse of its inverse (cid:103) U − d in a parallel quantum circuit is upperbounded by p max ≤ − d − k ( d − )+ d − .Proof. We construct our protocol by concatenating theprotocol for unitary complex conjugation of Ref. [25]with the unitary transposition one presented in Sec. VI B.First we divide the k uses of the input-operation (cid:102) U d into k (cid:48) = (cid:106) kd − (cid:107) groups containing d − (cid:102) U d and dis-card possible extra uses. We then exploit the unitaryconjugation protocol to obtain k (cid:48) uses of U ∗ d . After, weimplement the unitary transposition protocol of Sec. VI Bon k (cid:48) uses of (cid:102) U ∗ d to obtain a single use of (cid:103) U − d with prob-ability of success given by p = − d − k (cid:48) + d − .Next, we prove the upper bound. Let p inv ( d , k ) be thesuccess probability of transforming k uses of an arbit-rary unitary input-operation (cid:102) U d into a single use of itsinverse (cid:103) U − d with a parallel circuit. Suppose one has ac-cess to l = k ( d − ) uses of an input-operation (cid:102) U d . Onepossible protocol to transform these l uses of (cid:102) U d into itstranspose with a parallel circuit is the following, firstwe perform the deterministic parallel complex conjuga-tion protocol on l uses to obtain k uses of (cid:102) U ∗ d . We thenperform the parallel unitary inversion on k uses of (cid:102) U ∗ d to obtain (cid:93) U ∗ − d = (cid:102) U Td with probability p inv ( d , k ) . Thisparallel unitary transposition protocol has then successprobability of q T ( d , l ) = p inv ( d , k ) . Theorem 2 states thatany parallel circuit that transforms l uses of an arbitraryunitary into its transpose respects q T ( d , l ) ≤ − d − l + d − ,which implies p inv ( d , k ) ≤ − d − l + d − = − d − k ( d − ) + d − B. Adaptive unitary inversion circuit
For completeness we now summarise the protocol foradaptive unitary inversion presented in Ref. [29]. Ourprotocol to obtain (cid:103) U − d follows similar steps of the pro-tocol to implement (cid:103) U − d presented in the previous sectiongoes as follow (see Fig. 8).1. We start by making a d − U d toimplement the probabilistic heralded transpositionprotocol for unitary inverse used in Theorem 4 inthe main text. When the generalised Bell measure-ments return the outcomes i and j , the operation U d Figure 8. Flowchart illustrating the adaptive unitary inverseprotocol. is transformed into V = U − d X id Z jd , where X id and Z jd are the clock and shift operators, respectively(see Eq. (46)).2. If both outcomes i and j correspond to the iden-tity operator, i.e. , i = j =
0, we have V = U − d and we stop the protocol with success. If someother outcome is obtained, we make a single useof (cid:102) U d to apply X id − Z jd − U d into V and invert thetransformation of step 1 to obtain identity operator (cid:20) X id − Z jd − U d U − d Z jd X id = I d (cid:21) .3. Go to step 1.We see that step 1 requires d − (cid:103) U − d with probability (cid:16) − d (cid:17) . We need in total d uses of (cid:102) U d to complete steps 1 and 2. Iteration of this protocol leadsto a success probability of p = − (cid:16) − d (cid:17) (cid:98) k + d (cid:99) . C. Optimal protocols via SDP formulation and indefinitecausal order advantage
We now apply the SDP methods of Sec. III to the caseof unitary inversion and reproduce Table 1 of Ref. [29] inin Table II. For qubits ( d =
2) we note that, with the Pauliqubit unitary operator Y , YU Y = U ∗ for every U withdeterminant one. Hence, any protocol for transforminga single use of a qubit unitary operation into a singleuse of its transposition can be converted into a qubitunitary inversion protocol, vice versa . Hence the resultsand conclusions for qubits are equivalent to the onespresented in Sec. VI F.For qutrits ( d = d = k =
2, parallel, adapt-ive, and indefinite causal order protocols have attained d = k = = = = k = = ≈ ≈ k = = ≈ d = k = k = ≈ ≈ ≈ Table II. Maximum success probabilities for universally invert-ing k uses of U d , by parallel quantum circuit, adaptive quantumcircuit, and protocols with indefinite causal orders. Values inblue are analytical and in black via numerical SDP optimisation.This table is extracted from Table 1 of Ref. [29]. the same success probability, suggesting that our par-allel unitary inversion protocol may be optimal when d = k − VIII. CONCLUSIONS
We have addressed the problem of designing probabil-istic heralded universal quantum protocols that trans-form k uses of an arbitrary (possibly unknown) d -dimensional unitary quantum operation (cid:102) U d to exactlyimplement a single use of some other operation given by f ( (cid:102) U d ) . For the cases where f is a linear supermap, wehave provided a SDP algorithm that can be used to ana-lyse parallel, adaptive, and indefinite causal order proto-cols. For the parallel and adaptive cases, our algorithmfinds a quantum circuit that universally implements thedesired transformation with the optimal probability ofsuccess for any k and d . For the indefinite causal ordercase, the algorithm finds a quantum process that obtainsthe desired transformation with the optimal probabilityof success for any k and d .For the particular case of unitary complex conjugation, i.e. , f ( (cid:102) U d ) = (cid:102) U ∗ d we have proved that when k < d − k = d − (cid:102) U d into asingle use of its complex conjugation was presented inRef. [25], we can argue that the theoretical possibility ofimplementing universal exact unitary complex conjuga-tion is completely solved.For the particular case of unitary transposition, i.e. f ( (cid:102) U d ) = (cid:102) U Td , we have shown that the optimalsuccess probability with parallel circuits is exactly p = − d − k + d − . When adaptive circuits are considered,we have presented an explicit protocol that has successprobability p = − (cid:16) − d (cid:17) (cid:100) kd (cid:101) , which has an exponen-tial improvement over any parallel protocol. We havealso shown that indefinite causal order protocols out-0performs causally ordered ones by tackling the cases d = k ≤ d = k = i.e. f ( (cid:102) U d ) = (cid:103) U − d , we have proved that when k < d − k ≥ d − p = − (cid:16) − d (cid:17) (cid:98) k + d (cid:99) and we prove it to be exponentially higher than any success prob-ability obtained by parallel circuits. Acknowledgements: –
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