Generating a state t -design by diagonal quantum circuits
aa r X i v : . [ qu a n t - ph ] M a y Generating a state t -design by diagonal quantum circuits Yoshifumi Nakata, Masato Koashi, and Mio Murao Institute for Theoretical Physics, Leibniz University Hannover, GermanyDepartment of Physics, Graduate School of Science, University of Tokyo, Tokyo, Japan ∗ Photon Science Center, University of Tokyo, Tokyo, Japan Department of Physics, Graduate School of Science, University of Tokyo, Tokyo, JapanInstitute for Nano Quantum Information Electronics, University of Tokyo, Tokyo, Japan
We investigate protocols for generating a state t -design by using a fixed separable initial stateand a diagonal-unitary t -design in the computational basis, which is a t -design of an ensemble ofdiagonal unitary matrices with random phases as their eigenvalues. We first show that a diagonal-unitary t -design generates a O (1 / N )-approximate state t -design, where N is the number of qubits.We then discuss a way of improving the degree of approximation by exploiting non-diagonal gatesafter applying a diagonal-unitary t -design. We also show that it is necessary and sufficient to use O (log t )-qubit gates with random phases to generate a diagonal-unitary t -design by diagonal quan-tum circuits, and that each multi-qubit diagonal gate can be replaced by a sequence of multi-qubitcontrolled-phase-type gates with discrete-valued random phases. Finally, we analyze the number ofgates for implementing a diagonal-unitary t -design by non-diagonal two- and one-qubit gates. Ourresults provide a concrete application of diagonal quantum circuits in quantum informational tasks. I. INTRODUCTION
Diagonal quantum circuits in the computational basis are recently attracting much attention [1–5]. Despite thecommutativity of diagonal gates, it has been shown that they are likely to have stronger computational power thanclassical computers even if the initial state is fixed to be a separable state. This implies that the commutativity ofgates does not immediately result in a trivial computational power. From an experimental point of view, diagonalgates can be fault-tolerantly realized in, e.g., super- and semi-conducting systems [6]. Hence, any applications ofdiagonal circuits will lead to an experimental demonstration of a quantum advantage in informational tasks, but littleis known about concrete applications of diagonal circuits so far.One of the applications of diagonal circuits, proposed by two of the authors [7], is related to random states, whichare an ensemble of pure states uniformly distributed in a Hilbert space with respect to the unitarily invariant measure.They have many utilities in a wide range of applications, e.g., in quantum communicational tasks [8], for efficientmeasurements [9], for an algorithmic use [10, 11] and for estimation of gate fidelities [12]. Despite such applications,exact random states cannot be efficiently generated. Hence, efficient generations of a t -design of random states, calleda state t -design, using quantum circuits have been intensely studied [12–18, 20–23], where a t -design of an ensemble isan ensemble that simulates up to t th-order statistical moments of the original ensemble [9, 12, 19]. In most applicationsof random states, a state t -design for small t is sufficient [24] and it has been shown that an approximate state t -designcan be efficiently generated by a quantum circuit called a local random circuit [22]. In Ref. [7], a protocol has beenproposed for generating an exact state 2-design by combining a diagonal quantum circuit with a classical probabilisticprocedure, which provides a usage of diagonal quantum circuits that leads to several applications in quantum tasks.In this paper, we investigate protocols of generating a state t -design for general t by using a t -design of randomdiagonal-unitary matrices called a diagonal-unitary t -design . We first show that a good approximate state t -design isobtained simply by applying a diagonal-unitary t -design in the computational basis to a fixed separable state. Thedegree of approximation is given by t ( t − / N + O (1 / N ) for a constant t , where N is the number of qubits. Thisresult is interesting from two perspectives. From a theoretical point of view, it shows that a diagonal quantum circuitcan generate an ensemble of states whose distribution in a Hilbert space is hard to be distinguished from the uniformone as long as looking at lower order statistical moments. This may help an intuitive understanding of a strongcomputational power of diagonal quantum circuits. On the other hand, from experimental point of view, our protocolextends a usage of diagonal quantum circuits in quantum applications and can be used for demonstrating a quantumadvantage.We also study a way of improving the degree of approximation by using a local random circuit after applying adiagonal-unitary t -design. Since an ensemble of states after a diagonal-unitary t -design is already a good approximate ∗ Electronic address: [email protected] state t -design, it is natural to expect that this protocol has an advantage to reduce the number of gates in the localrandom circuit compared to the one that uses only a local random circuit. We numerically confirm that this seemsto be the case.It is also important to investigate efficient implementations of a diagonal-unitary t -design by quantum circuits. Al-though a diagonal-unitary t -design contains only diagonal matrices, it cannot be implemented in general by using onlytwo- and one-qubit diagonal gates since multi-qubit diagonal gates are generally not decomposable into diagonal gatesacting on smaller number of qubits. For instance, a three-qubit gate diag(1 , , , , , , , −
1) cannot be representedby a quantum circuit consisting of only two- and one-qubit diagonal gates. This indecomposability of multi-qubitdiagonal gates into two-qubit diagonal gates should be contrasted with the decomposability of multi-qubit generalunitary gates into two- and one-qubit unitary gates. We show that, if we use only diagonal gates, it is necessary andsufficient to use ( ⌊ log t ⌋ + 1)-qubit gates with random phases for generating a diagonal-unitary t -design, where ⌊ x ⌋ denotes the maximum integer that does not exceed x . We also show that the multi-qubit diagonal gates in the circuitfor implementing a diagonal-unitary t -design can be replaced by multi-qubit controlled-phase-type gates with discrete random phases. We finally discuss how to generate a diagonal-unitary t -design by using non-diagonal two-qubit gates,and provide a construction of a quantum circuit implementing a diagonal-unitary t -design by O ( N log t ) two-qubitgates for a constant t , while it will not be optimal.Before leaving the introduction, we would like to note that partially randomizing unitary matrices while preservingsome properties, which is the case in a diagonal-unitary t -design, is not necessarily simpler than full randomizationin the unitary group, which is a concern of a unitary t -design. For a partial randomization, we need to performtwo conflict tasks, randomization and preservation , at the same time. Thus, although commutativity of diagonalgates simplifies an investigation of random diagonal-unitary matrices, it is not trivial whether an implementation ofdiagonal-unitary t -design is simpler than that of a unitary t -design.This paper is organized as follows. In Sec. II, we review the definitions of terms used in this paper. We summarizeall of our main results in Sec. III. Their proofs are provided in Sec. IV. We make concluding remarks in Sec. V. II. RANDOM UNITARY MATRICES AND t -DESIGNS We first review the definitions of random unitary and diagonal-unitary matrices [7, 25], random and phase-randomstates [26], and their t -designs [19]. In the following, we denote by | i and | i the computational basis of the Hilbertspace of a qubit, which are the eigenstates of the Pauli Z operator with eigenvalues +1 and −
1, respectively. Forsimplicity, we also denote by E expectations over a probability distribution. If necessary, we specify the probabilityspace taken over for the expectation. Definition 1 (Random unitary matrices and random states)
Let U ( d ) be the unitary group of degree d . Ran-dom unitary matrices U Haar are the ensemble of unitary matrices uniformly distributed with respect to the Haarmeasure on U ( d ). Random states Υ Haar are the ensemble of states { U | Ψ i} U ∈U Haar for any fixed state | Ψ i ∈ H , where H is a Hilbert space with dimension d . Definition 2 (Random diagonal-unitary matrices and phase-random states)
Random diagonal-unitary ma-trices in an orthonormal basis {| u n i} denoted by U diag ( {| u n i} ) are an ensemble of diagonal unitary matrices of theform U ϕ = P dn =1 e iϕ n | u n ih u n | , where the phases ϕ n are uniformly distributed according to the normalized Lebesguemeasure d ϕ = d ϕ · · · d ϕ d / (2 π ) d on [0 , π ) d . Phase-random states Υ phase ( {|h u n | Ψ i| , | u n i} ) are an ensemble of states { U | Ψ i} U ∈U diag ( {| u n i} ) .Note that a distribution of random states is independent of an initial state | Ψ i due to the unitary invariance of theHaar measure. This is not the case in phase-random states and their distribution depends on the initial state.A t -design of an ensemble is defined by an ensemble that simulates up to the t th-order statistical moments ofthe original ensemble on average [9, 12, 19, 27]. Although a t -design is required to be a finite ensemble in severaldefinitions, we do not require it to be more general. Definition 3 ( ǫ -approximate unitary t -designs) Let U be random unitary matrices or random diagonal-unitarymatrices. An ǫ -approximate t -design of U , denoted by U ( t,ǫ ) , is an ensemble of unitary matrices such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E U ∈U ( t,ǫ ) [ U ⊗ t ⊗ ( U † ) ⊗ t ] − E U ∈U [ U ⊗ t ⊗ ( U † ) ⊗ t ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ, where || · || = tr | · | is the trace norm. The t -designs for random unitary and diagonal-unitary matrices are called unitary and diagonal-unitary t -designs, respectively. Definition 4 ( ǫ -approximate state t -designs) Let Υ be random states or phase-random states. An ǫ -approximate t -design of Υ, denoted by Υ ( t,ǫ ) , is an ensemble of states such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E | ψ i∈ Υ ( t,ǫ ) [ | ψ ih ψ | ⊗ t ] − E | ψ i∈ Υ [ | ψ ih ψ | ⊗ t ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ. In particular, we call a t -design of random states a state t -design in this paper.A t -design for ǫ = 0 is called an exact t -design. We mean the exact ones when we simply use the term t -designsin this paper. Although we have presented definitions of ǫ -approximate t -designs in terms of the trace norm, thereare other definitions using different distance measures such as the diamond norm and the Schatten norms (see, e.g.,Ref. [28]). However, they are shown to be all equivalent,namely, if V is an ǫ -approximate t -design in one of thedefinitions, then it is also an ǫ ′ -approximate t -design in other definitions, where ǫ ′ = poly(2 tN ) ǫ [28].A unitary and a state t -design can be used in many quantum informational tasks. For instance, random statessaturate the classical communication capacity of a noisy quantum channel [8], and are also related to optimal measure-ments in tomography [9]. POVM measurements in a random basis can be used for solving hidden subgroup problemsefficiently [10]. Random states in these use can be replaced by a state t -design for a small t [24]. A 2-design of randomstates is known to be useful for checking the fidelity of quantum gates [12]. III. MAIN RESULTS
We summarize our results in this section. In Subsec. III A, we provide protocols of generating a state t -design byusing a diagonal-unitary t -design in the computational basis. Our results about implementations of diagonal-unitary t -designs by diagonal circuits are presented in Subsec. III B. All the proofs are given in Section 4. A. Protocols of generating a state t -design by using a diagonal-unitary t -design Applying a diagonal-unitary t -design on any pure state achieves a t -design of the corresponding phase-randomstates by definition. If we choose an appropriate initial state and a basis of the diagonal-unitary t -design, we can alsoachieve a good approximation of a t -design of random states , as stated in the following Proposition: Proposition 1 A t -design of phase-random states obtained by applying a diagonal-unitary t -design in the computa-tional basis onto an initial state | + i ⊗ N , where | + i = √ ( | i + | i ), is an η ( N, t )-approximate state t -design, where η ( N, t ) = t ( t − d + O ( d ) and d = 2 N .As we will see in the next subsection, a diagonal-unitary t -design for a small t can be achieved by using only diagonalgates acting on a small number of qubits, where the order of the applications of gates does not matter due to thecommutativity of diagonal gates, i.e., there is no inherent temporal structure in the circuit. This is a big advantagein experimental implementations of the circuit. In particular, a diagonal-unitary t -design for t ≤ O ( N ). This means that the protocolin Proposition 1 generates an η ( N, t )-approximate state t -design for t ≤ O ( N )two- and one-qubit diagonal gates that has no temporal structure. This should be contrasted to a previously knownprotocol using a local random circuit [22], which is composed of two-qubit gates randomly chosen from U (4) actingonly on neighboring qubits. Although it achieves the same degree of approximation by using at most O ( t log( t ) N )gates, the circuit is necessarily temporally structured. Thus, our protocol has a practical advantage for large N aslong as the required degree of approximation is up to η ( N, t ), particularly when t ≤ t -design with other proce-dures. In Ref. [7], it was shown that an exact state 2-design is obtained if we combine a diagonal-unitary 2-design witha classical probabilistic procedure. However, we can show that adding a classical probabilistic procedure improvesthe degree of approximation by O ( d − t ) in general, so that it is not effective for t ≥ t -design on | + i ⊗ N . Sincean η ( N, t )-approximate state t -design is already achieved by a diagonal-unitary t -design, it is natural to expect thatthis protocol generates an ǫ -approximate state t -design more efficiently than the one that uses only a local randomcircuit. We numerically check this. In contrast to the previous results about a local random circuit, where an inputstate is arbitrary, input states of the local random circuit in our protocol are determined by output states obtained ç ç ç ç ç ç ç ç ç ç ç ç ç ç çá á á á á á á á á á á á á á áó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ - T r ace D i s t a n ce , D H T L æ æ æ æ - - - - Α FIG. 1: The left figure shows how D ( T ) changes when we apply a parallelized local random circuit of length T after theapplication of a diagonal-unitary t -design on the initial state | + i ⊗ N . The numerics is performed by Mathematica 9 for t = 2and each plot ◦ (green), (cid:3) (red), △ (purple), and × (blue), represents N = 3 , , ,
6, respectively. The number of sampling ateach point of the plot is 1000. The right figure shows how the coefficient α depends on N .FIG. 2: An r -qubit phase-random circuit for an N -qubit system where N = 5 and r = 3. The black circles imply theplaces where the r -qubit gate acts on. Each r -qubit gate is diagonal in the computational basis with random phases, i.e.,diag( e iφ , e iφ , · · · , e iφ r ) and φ k ∈ [0 , π ). Note that the number of r -qubit diagonal gates is given by (cid:0) Nr (cid:1) . by applying a diagonal-unitary t -design on | + i ⊗ N . Hence, the necessary length of the local random circuit in ourprotocol is not directly obtained from the previous results.Let T be the length of a parallelized local random circuit after applying a diagonal-unitary t -design, where we meanby a parallelized local random circuit that unitary gates acting on different qubits are applied simultaneously. Wedenote by D ( T ) the trace distance between an expectation of | φ ih φ | ⊗ t over the resulting ensemble and that over astate t -design (see Definition 4). We numerically check how D ( T ) scales with T for t = 2 and N = 3 , , ,
6. In thenumerics, we randomly generate a unitary matrix representing a parallelized local random circuit, and apply it tothe states obtained by applying a diagonal-unitary t -design on | + i ⊗ N . By repeating this and averaging the resultingstates, we evaluate the expectation of | φ ih φ | ⊗ t over the ensemble obtained by our protocol. The result is shown inFig. 1. We observe that the distance exponentially decreases for each N . Although the exponent of the exponentialdecrement depends on N for up to N = 6, we expect from the right figure in Fig. 1 that it converges to some value α ( t ) that depends on t but not on N . Accordingly, we conjecture the following: Conjecture 1
Let D ( T ) be the trace distance as defined above. Then, D ( T ) ∼ η ( N, t )2 − T/α ( t ) ∼ t ( t − d − T/α ( t ) . If this is the case, a parallelized random circuit of length T ( ǫ ) following a diagonal-unitary t -design achieves an ǫ -approximate state t -design, where T ( ǫ ) = ( ǫ ≥ η ( N, t ) ,α ( t )[log /ǫ − N + log t ( t − ǫ < η ( N, t ) . B. Implementations of diagonal-unitary t -designs We present our result that an r -qubit phase-random circuit achieves a diagonal-unitary t -design, where r is de-termined by t . An r -qubit phase-random circuit is an extension of a phase-random circuit [7, 26]. An r -qubitphase-random circuit for an N -qubit system is a quantum circuit consisting of r -qubit diagonal gates in the computa-tional basis with random phases applied on all combinations of r qubits out of N qubits (see also Fig. 2). Note thateach r -qubit gate cannot be decomposed into a sequence of s -qubit diagonal gates ( s < r ) since the phases of r -qubitdiagonal gates should be chosen independently and randomly, which cannot be achieved by randomizing the phasesof gates acting only on s < r qubits.Our first result on an implementation of diagonal-unitary t -designs is given in the following theorem. Theorem 1 An r -qubit phase-random circuit is a diagonal-unitary t -design if and only if r ≥ ⌊ log t ⌋ +1 for t ≤ N − r = N for t ≥ N .This result implies that we cannot achieve a diagonal-unitary t -design for t ≥ r -qubit gate for implementing a diagonal-unitary t -design can be replaced by asequence of multi-qubit controlled-phase-type gates that act on s qubits ( s ≤ r ), where a multi-qubit controlled-phase-type gate is a unitary operation represented by diag(1 , , · · · , , e iα ) in the computational basis. We also provethat a phase of a multi-qubit controlled-phase gate acting on s qubits can be randomly selected from a ( ⌊ t/ s − ⌋ + 1)-valued discrete set of phases. Thus, the smaller number of discrete phases is required for the gates acting on thelarger number of qubits. This result also shows that a phase-random circuit achieves a diagonal-unitary t -design witha finite number of elements.These results enable us to analyze implementations of diagonal-unitary t -designs by using two- and one-qubit non-diagonal gates. An explicit construction of a multi-qubit controlled-phase-type gate acting on r qubits is known and itrequires O ( r ) two-qubit gates [29], although it is unlikely to be optimal. By decomposing the multi-qubit controlled-phase gates in an r -qubit phase-random circuit, we can show that a diagonal-unitary t -design for an N -qubit systemis obtained after applying M two-qubit non-diagonal gates, where M = (cid:18) Nr (cid:19) r − X s =1 O ( s ) (cid:18) rs (cid:19) , and r is given in Theorem 1. For a constant t , M ∼ O ( N log t ) and the construction is efficient. For larger t such as t = poly( N ), this provides a sub-exponential implementation of a diagonal-unitary t -design. IV. PROOFS
We present proofs of all statements presented in Sec. III. In Subsec. IV A, we show the proof of Proposition 1. Wepresent implementations of a diagonal-unitary t -design by an r -qubit phase-random circuit in Subsec. IV B and showa decomposition of each r -qubit gate into multi-qubit controlled-phase gates in Subsec. IV C.Before presenting the proofs, we introduce our notation. We denote t N -bit sequences by n := ( ~n (1) , · · · , ~n ( t ) ), where ~n ( k ) is an N -bit sequence for k = 1 , · · · , t , and a set of all n by D , i.e., D = { ( ~n (1) , · · · , ~n ( t ) ) | ~n ( k ) ∈ { , } × N for k =1 , · · · , t } . Let P t be a permutation group of order t . We introduce an equivalent relation in D by P t ; n ∼ m ifand only if there exists σ ∈ P t such that ~n ( i ) = ~m ( σ ( i )) for all i ∈ { , · · · , t } . We denote by D / P t a quotient set of D by the equivalent relation. For simplicity, we choose a representative of each equivalent class by n that satisfies ~n ( i ) ≤ ~n ( i +1) for every i ∈ { , · · · , t − } . Note that the inequality is taken in binary, namely, ~n ( i ) ≤ ~n ( j ) if andonly if P Nk =1 ~n ( i ) k N − k ≤ P Nk =1 ~n ( j ) k N − k , where ~n ( i ) k ∈ { , } is the k th bit of ~n ( i ) . We finally introduce a canonicalmap π from D to D / P t , and define π − ( n ′ ) = { n ∈ D| π ( n ) = n ′ } for n ′ ∈ D / P t . Using this notation, we define for n ′ ∈ D / P t , (cid:12)(cid:12) π − ( n ′ ) (cid:11) := 1 p | π − ( n ′ ) | X m ∈ π − ( n ′ ) | m i , where | π − ( n ′ ) | is the number of elements in π − ( n ′ ). A. Generating an approximate state t -design by a diagonal-unitary t -design We show Propositions 1 given in Sec. III.
Proof 1
The expectation of | φ ih φ | ⊗ t over random states is given by E | φ i∈ Υ Haar [ | φ ih φ | ⊗ t ] = 1 (cid:0) t + d − t (cid:1) X n ′ ∈D / P t (cid:12)(cid:12) π − ( n ′ ) ih π − ( n ′ ) (cid:12)(cid:12) . This is obtained by simply applying Schur’s lemma [30]. For a t -design of phase-random states Υ phase obtained byapplying a diagonal-unitary t -design on | + i ⊗ N , the expectation is given by E | φ i∈ Υ phase [ | φ ih φ | ⊗ t ] = 1 d t X n ′ ∈D / P t | π − ( n ′ ) | (cid:12)(cid:12) π − ( n ′ ) ih π − ( n ′ ) (cid:12)(cid:12) . The difference η ( N, t ) between the expectations is given by η ( N, t ) = || E | φ i∈ Υ Haar [ | φ ih φ | ⊗ t ] − E | φ i∈ Υ phase [ | φ ih φ | ⊗ t ] || = X n ′ ∈D / P t (cid:12)(cid:12)(cid:12)(cid:12) (cid:0) t + d − t (cid:1) − d t | π − ( n ′ ) | (cid:12)(cid:12)(cid:12)(cid:12) . We expand (cid:0) t + d − t (cid:1) by t ! P tk =1 α k d k , where α t = 1 and α t − = t ( t − /
2, and obtain η ( N, t ) = (cid:18) t X k =1 α k d k − t (cid:19) − X n ′ ∈D / P t (cid:12)(cid:12)(cid:12)(cid:12) t ! 1 d t − | π − ( n ′ ) | t X k =1 α k d t − k (cid:12)(cid:12)(cid:12)(cid:12) . (1)We explicitly calculate this up to the order 1 /d . For d ≫ t , ( P tk =1 α k d k − t ) − is given by (cid:18) t X k =1 α k d k − t (cid:19) − = 1 − t ( t − d + O ( 1 d ) . (2)The other term in Eq. (1), | π − ( n ′ ) | , depends only on the number of the same N -bit sequences in n ′ . For n ′ suchthat ~n ( i ) = ~n ( j ) for i = j , | π − ( n ′ ) | = t ! and the number of such n ′ is (cid:0) dt (cid:1) . For n ′ such that there exists only one pair( i, j ), where i = j , satisfying ~n ( i ) = ~n ( j ) , | π − ( n ′ ) | = t ! / n ′ is ( t − (cid:0) dt − (cid:1) . In other cases,the number of each type of n ′ is at most d t − . Since the inside of the summation P n ′ ∈D / P t in Eq. (1) is at most O (1 /d t ), we obtain that X n ′ ∈D / P t (cid:12)(cid:12)(cid:12)(cid:12) t ! 1 d t − | π − ( n ′ ) | t X k =1 α k d t − k (cid:12)(cid:12)(cid:12)(cid:12) = (cid:18) dt (cid:19)(cid:18) t ! t ( t − d t +1 + O ( 1 d t +2 ) (cid:19) + ( t − (cid:18) dt − (cid:19)(cid:18) t !2 1 d t + O ( 1 d t +1 ) (cid:19) + O ( 1 d )= t ( t − d + O ( 1 d ) , (3)where we have used relations such as (cid:0) dt (cid:1) = ( d t + O ( d t − )) /t ! for a constant t . From Eqs. (2) and (3), we obtain η ( N, t ) = t ( t − d + O ( 1 d ) , (cid:4) B. An r -qubit phase-random circuit achieves a diagonal-unitary t -design We prove Theorem 1 by restating it in a different way. Since the statement is obvious for r = N , we consider only r ≤ N − S be E U ∈U diag [ U ⊗ t ⊗ ( U † ) ⊗ t ] and introduce expansion coefficients S nm in the computational basis defined by S = X n , m ∈D S nm | n ih n | ⊗ | m ih m | . Note that S is diagonal in the computational basis since S is an expectation of U ⊗ t ⊗ ( U † ) ⊗ t for U ∈ U diag , and U ∈ U diag is diagonal in the computational basis. A simple calculation leads to S nm = ( π ( n ) = π ( m ) , π ( n ) = π ( m ) . (4)Our goal is to derive the value of r for which the r -qubit phase-random circuit achieves the coefficients S nm .We denote by I s a subset of { , · · · , N } with s elements. For a given I s = { i , · · · , i s } , we denote s -bit subse-quences in ~n ( k ) and ~m ( k ) at I s by ~n ( k ) I s := n ( k ) i · · · n ( k ) i s and ~m ( k ) I s := m ( k ) i · · · m ( k ) i s , respectively, and ( ~n (1) I s , · · · , ~n ( t ) I s ) and( ~m (1) I s , · · · , ~m ( t ) I s ) by n I s and m I s , respectively. We generalize a canonical map π to the one mapping t s -bit sequences D s to a quotient set D s / P t by the permutation group P t . In this notation, D N = D . We call the number of 1 in a bitsequence weight of the sequence. By using these expressions, we first prove the following lemma. Lemma 1
Let n , m ∈ D s be such that π ( n ) = π ( m ) and π ( n I s − ) = π ( m I s − ) for any I s − ⊂ { , · · · , s } . Denote by G n ( ~n ) the number of ~n in n . Then, for any s -bit sequence ~n , | G n ( ~n ) − G m ( ~n ) | = g, (5)where g is a constant positive integer. Moreover, t ≥ s − g . Proof 2 If ∃ q, q ′ ∈ { , · · · , t } such that ~n ( q ) = ~m ( q ′ ) , we remove them from n and m . As a result, we obtain ˜ n and˜ m , which are composed of t ′ s -bit sequences for t ′ ≤ t . Note that ˜ n and ˜ m still satisfy π (˜ n I s − ) = π ( ˜ m I s − ) for any I s − ⊂ { , · · · , s } .Without loss of generality, we assume that the s -bit sequence with the most occurrence in ˜ n is 00 · · ·
0, and let g be the number of the occurrence. Since π (˜ n I s − ) = π ( ˜ m I s − ) for any I s − ⊂ { , · · · , s } and all s -bit sequences in˜ n differ from those in ˜ m , all sequences with weight one should be contained in ˜ m . Moreover, the number of eachsequence with weight one in ˜ m is g since the number of 00 · · · n is g . This in turn implies that all sequences withweight two should be contained in ˜ n . Similarly, the number of each of such sequences should be g . By repeating this,it follows that all sequences with zero or even weight are contained in ˜ n , and those with odd weight are in ˜ m . Inaddition, the number of each sequence is g . Thus, we obtain for any s -bit sequence ~n , | G ˜ n ( ~n ) − G ˜ m ( ~n ) | = g, and t ′ = 2 s − g . By construction, | G ˜ n ( ~n ) − G ˜ m ( ~n ) | = | G n ( ~n ) − G m ( ~n ) | , so that we obtain Eq. (5). Moreover, as t ≥ t ′ ,it follows that t ≥ s − g . (cid:4) By using Lemma 1, we show the following Proposition.
Proposition 2
For r ≤ N −
1, the following are equivalent;(A) An r -qubit phase-random circuit achieves an exact diagonal-unitary t -design,(B) For any n , m ∈ D , if π ( n I r ) = π ( m I r ) for any I r ⊂ { , · · · , N } , then π ( n ) = π ( m ),(C) r > log t .From the equivalence of (A) and (C), we obtain Theorem 1. Proof 3
We first show the equivalence of (A) and (B), and then that of (B) and (C). The unitary matrix correspondingto an r -qubit phase-random circuit is given by W φ = Y I r ⊂{ , ··· ,N } W I r , where W I r := diag I r ( e iφ , · · · , e iφ r ) ⊗ I { , ··· ,N }\ I r is a diagonal unitary matrix with random phases { φ , · · · , φ r } acting non-trivially on the qubits at sites I r . The matrix I I represents the identity matrix acting on qubits at sites I ⊂ { , · · · , N } . Since the random phases are independently chosen for each W I r , the expectation of W ⊗ tφ ⊗ ( W † φ ) ⊗ t over all random phases is given by E [ W ⊗ tφ ⊗ ( W † φ ) ⊗ t ] = Y I r ⊂{ , ··· ,N } E [ W ⊗ tI r ⊗ ( W † I r ) ⊗ t ]= Y I r ⊂{ , ··· ,N } X n , m ∈D W nm I r | n ih n | ⊗ | m ih m | , where W nm I r = ( π ( n I r ) = π ( m I r ) , π ( n I r ) = π ( m I r ) . (6)By comparing Eq. (6) with Eq. (4), we obtain the equivalence of (A) and (B).Next, we show that (C) implies (B) by showing its contraposition, namely, if there exists n , m ∈ D that satisfies π ( n ) = π ( m ) but ∀ I r ⊂ { , · · · , N } , π ( n I r ) = π ( m I r ), then t ≥ r . By assumption, there exists r ′ ≥ r and I r ′ +1 suchthat π ( n I r ′ +1 ) = π ( m I r ′ +1 ) and π ( n I r ′ ) = π ( m I r ′ ) for any I r ′ ⊂ I r ′ +1 . It follows from Lemma 1 that t ≥ r ′ g , where g ≥
1. As r ′ ≥ r , we obtain t ≥ r .Finally, we show that (B) implies (C). This is also obtained by showing its contraposition. Consider n and m ∈ D such that, for a fixed I r +1 , n I r +1 and m I r +1 contain all ( r + 1)-bit sequences with even or zero weight and thosewith odd weight, respectively, and n { , ··· ,N }\ I r +1 = m { , ··· ,N }\ I r +1 . Such n and m exist if t ≥ r . It is obvious that π ( n ) = π ( m ). However, it is easy to see that π ( n I r ) = π ( m I r ) for any I r ⊂ { , · · · , N } . This shows the contrapositionof the statement that (B) implies (C), and concludes the proof. (cid:4) C. Decomposition of r -qubit gates into the controlled-phase-type gates We show how to decompose each r -qubit diagonal gate in an r -qubit phase-random circuit into a sequence ofmulti-qubit controlled-phase-type gates with discrete random phases. More precisely, we prove the following. Proposition 3
To implement a diagonal-unitary t -design by an r -qubit phase-random circuit, every gatediag I r (1 , e iφ , · · · , e iφ r − ) ⊗ I { , ··· ,N }\ I r non-trivially acting on r qubits at I r with random phases φ k ∈ [0 , π ) for k = 1 , · · · , r − D I r = Y I s ⊂ I r , ≤ s ≤ r diag I s (1 , , · · · , , e iα Is ) ⊗ I { , ··· ,N }\ I s , where α I s are randomly and independently chosen from (cid:26) πk ⌊ t/ s − ⌋ + 1 (cid:27) k =0 , , ··· , ⌊ t/ s − ⌋ . (7) Proof 4
We consider an r -qubit diagonal gate acting on qubits at I r . Since it is sufficient to consider a nontrivialpart of the matrix, we investigate a diagonal matrix given by W I r = X ~n Ir e iφ ~nIr | ~n I r ih ~n I r | . Its tensor product W ⊗ t Ir ⊗ ( W † Ir ) ⊗ t is given by W ⊗ tI r ⊗ ( W † I r ) ⊗ t = X n Ir , m Ir ∈D r e iφ n Ir m Ir | n I r ih n I r | ⊗ | m I r ih m I r | , where φ n Ir m Ir = P tk =1 ( φ ~n ( k ) Ir − φ ~m ( k ) Ir ). If every phase is randomly chosen from [0 , π ), E [ e iφ n Ir m Ir ] = ( π ( n I r ) = π ( m I r ) , π ( n I r ) = π ( m I r ) . (8)We investigate the coefficient of | n I r ih n I r | ⊗ | m I r ih m I r | in D ⊗ tI r ⊗ ( D † I r ) ⊗ t . Our goal is to show that the choiceof the phases defined in Eq. (7) achieves the same average as Eq. (8). For this purpose, it is sufficient to prove thefollowing two properties: (a) For any I ⊂ I r , the average over α I makes the coefficient of | n I ih n I | ⊗ | m I ih m I | vanish if π ( n I ) = π ( m I ). (b) For any I s ⊂ I r with any 1 < s ≤ r , the average over α I s makes the coefficients of | n I s ih n I s | ⊗ | m I s ih m I s | vanish if π ( n I s ) = π ( m I s ) and π ( n I s − ) = π ( m I s − ) for all I s − ⊂ I s .The property (a) holds since the coefficients of | n I ih n I | ⊗ | m I ih m I | in D ⊗ tI r ⊗ ( D † I r ) ⊗ t includes the factor e it ′ α I with 1 ≤ t ′ ≤ t , which vanishes after taking the average of α I over { πkt +1 } k =0 , , ··· ,t . The property (b) is obtained fromLemma 1. When n I s and m I s satisfy π ( n I s ) = π ( m I s ) and π ( n I s − ) = π ( m I s − ) for all I s − ⊂ I s , it follows fromLemma 1 that, for an s -bit sequence ~ s := 11 · · · | G n Is ( ~ s ) − G m Is ( ~ s ) | = g. This implies that the coefficient of | n I s ih n I s | ⊗ | m I s ih m I s | for such n I s and m I s contains e igα Is or e − igα Is . We alsoobtain from Lemma 1 that t ≥ s − g , namely, g ≤ ⌊ t/ s − ⌋ , where the equality holds for n I s and m I s that do notshare the same s -bit sequences. Thus, for these terms to be zero by taking the average over α I s , it is sufficient to take α I s randomly and independently from (cid:26) πk ⌊ t/ s − ⌋ + 1 (cid:27) k =0 , ··· , ⌊ t/ s − ⌋ . (cid:4) V. SUMMARY AND CONCLUDING REMARKS
We investigated protocols of generating a state t -design in an N -qubit system by using a diagonal-unitary t -designin the computational basis applied on a fixed separable state. We have first shown that a O (1 / N )-approximate state t -design is generated by simply applying the diagonal-unitary t -design. We have then investigated a way of improvingthe degree of approximation by exploiting a local random circuit in addition to a diagonal-unitary t -design, whichseems to result in a faster convergence than the protocol using only a local random circuit. We have also investigatedquantum circuit implementations of a diagonal-unitary t -design, and have shown that an r -qubit phase-random circuit,where r ≥ ⌊ log t ⌋ + 1 for t ≤ N − r = N for t ≥ N , generates a diagonal-unitary t -design. The number of r -qubit gates in the circuit is given by (cid:0) Nr (cid:1) . Each r -qubit diagonal gate has been shown to be decomposable into asequence of s -qubit multi-qubit controlled-phase gates ( s ≤ r ) with ( ⌊ t/ s − ⌋ + 1)-valued discrete random phases.We make remarks on possible future directions. First, numerical analysis of the method applying a local randomcircuit in addition to a diagonal-unitary t -design is less conclusive, so that further numerical or analytical investigationsare required. For an analytical investigation, it is sufficient to check how the coefficients of (cid:12)(cid:12) π − ( n ) ih π − ( n ) (cid:12)(cid:12) arechanged by a local random circuit. Although we studied a protocol of generating a state t -design from a fixed initialstate in this paper, it is interesting to investigate if our protocol also achieves a unitary t -design more efficiently.Another direction is to deepen the analysis of a quantum circuit implementation of a diagonal-unitary t -designby using non-diagonal gates. We have provided an implementation of an exact diagonal-unitary t -design by using O ( N log t ) non-diagonal two-qubit gates for a constant t . However, the scaling is worse for a large t than that of aunitary t -design implemented by a local random circuit, which requires O ( N t ( N + log 1 /ǫ )) non-diagonal two-qubitgates [22]. Since our implementation is probably not optimal, it is interesting to see a lower bound of the length ofthe circuit to implement a diagonal-unitary t -design.It will be also interesting to investigate in which cases a unitary t -design used in a quantum protocol or taskcan be subsituted by a diagonal-unitary t -design. We have shown in this paper that generation of a state t -design0is one of the cases. Random unitary matrices have been also exploited for decoupling two systems [31, 32]. Sincedecoupling has many applications in quantum information processing, it may be interesting and useful to investigateif diagonal-unitary t -designs are capable to achieve an exact or approximate decoupling. VI. ACKNOWLEDGMENT
This work is supported by Project for Developing Innovation Systems of the Ministry of Education, Culture, Sports,Science and Technology (MEXT), Japan. Y. N. and M. M acknowledge support from JSPS by KAKENHI, GrantNo. 222812 and 23540463, respectively. Y. N. also acknowledges JSPS Postdoctoral Fellowships for Research Abroad.M. K is supported by the Funding Program for World-Leading Innovative R&D on Science and Technology (FIRST).
Appendix A: Generating a state t -design by a diagonal-unitary t -design and a probabilistic procedure We consider a protocol of generating a state t -design by using a diagonal-unitary t -design and a classical probabilisticprocedure, and show that the improvement of the degree of approximation is limited to be O ( d − t ).We generalize a protocol introduced in Ref. [7] as follows:1. With probability p , apply a diagonal-unitary t -design on | + i ⊗ N .2. With probability 1 − p , choose a random N -bit sequence ~n and generate | ~n i .We denote by Υ( p ) the resulting ensemble and show thatmin p ∈ [0 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E | ψ i∈ Υ Haar [ | ψ ih ψ | ⊗ t ] − E | ψ i∈ Υ( p ) [ | ψ ih ψ | ⊗ t ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = η ( N, t ) − O ( 1 d t − ) , where the minimum is given by p = 1 − d/ (cid:0) t + d − t (cid:1) − d − t . Following the calculations in Subsec. IV A, the difference between the expectation over random states and that overΥ( p ) is given by D ( p ) := || E | ψ i∈ Υ Haar [ | φ ih φ | ⊗ t ] − E | φ i∈ Υ( p ) [ | φ ih φ | ⊗ t ] || = d (cid:12)(cid:12)(cid:12)(cid:12) (cid:0) t + d − t (cid:1) − ( 1 − pd + pd t ) (cid:12)(cid:12)(cid:12)(cid:12) + X n ′ ∈D ′ / P t (cid:12)(cid:12)(cid:12)(cid:12) (cid:0) t + d − t (cid:1) − pd t | π − ( n ′ ) | (cid:12)(cid:12)(cid:12)(cid:12) , where D ′ = D \ { ( ~n, ~n, · · · , ~n ) | ~n ∈ { , } × N } . In this notation, D (1) = η ( N, t ).Since D ( p ) is a linear function of p , it is sufficient to investigate the coefficient of p . It is straightforward to observethe following: 1 (cid:0) t + d − t (cid:1) − ( 1 − pd + pd t ) < p < − d/ ( t + d − t ) − d − t ,> p > − d/ ( t + d − t ) − d − t , for n ′ = ( ~n (1) , · · · , ~n ( t ) ), where ~n ( i ) = ~n ( j ) for i = j ,1 (cid:0) t + d − t (cid:1) − pd t | π − ( n ′ ) | > p < d t ( t + d − t ) t ! ,< p > d t ( t + d − t ) t ! , and, for other n ′ , 1 (cid:0) t + d − t (cid:1) − pd t | π − ( n ′ ) | > . p is negative for p < p and positivefor p > p , where p = − d/ ( t + d − t ) − d − t , so that D ( p ) is the minimum. The order of D ( p ) is easily estimated from afact that p ∼ − ( t ! + 1) d − t . Since p is a probability of mixing a t -design of Υ(1) and a separable state {| ~n ih ~n |} , D (1) − D ( p ) = O ( d − t ), resulting in min p D ( p ) = η ( N, t ) − O ( d − t ). [1] D. Shepherd and M. J. Bremner, Proc. R. Soc. A 465, 1413-1439 (2009).[2] M. J. Bremner and R. Jozsa and D. J. Shepherd, Proc. R. Soc. A 8 vol. 467 no. 2126 459-472 (2011).[3] X. Ni and M. van den Nest, Quantum Information and Computation, Vol. 13, No. 1-2 0054-0072 (2013).[4] M. J. Hoban, J. J. Wallman, H. Anwar, N. Usher, R. Raussendorf, D. E. Browne, Phys. Rev. Lett. (2014) 140505.[5] K. Fujii and T. Morimae, arXiv:1311.2128 (2013).[6] P. Aliferis, F. Brito, D. P. DiVincenzo, J. Preskill, M. Steffen and B. M. Terhal, New J. Physics (2009) 013061.[7] Y. Nakata and M. Murao, Int. J. Quantum Inform. (1997) 1613.[9] J. M. Renes, R. Blume-Kohout, A. J. Scott and C. M. Caves, J. Math. Phys.