Pretty good state transfer via adaptive quantum error correction
PPretty good state transfer via adaptive quantum error correction
Akshaya Jayashankar and Prabha Mandayam Department of Physics, Indian Institute of Technology Madras, Chennai, India 600036
We examine the role of quantum error correction (QEC) in achieving pretty good quantum statetransfer over a class of 1-d spin Hamiltonians. Recasting the problem of state transfer as oneof information transmission over an underlying quantum channel, we identify an adaptive QECprotocol that achieves pretty good state transfer. Using an adaptive recovery and approximateQEC code, we obtain explicit analytical and numerical results for the fidelity of transfer over idealand disordered 1-d Heisenberg chains. In the case of a disordered chain, we study the distributionof the transition amplitude, which in turn quantifies the stochastic noise in the underlying quantumchannel. Our analysis helps us to suitably modify the QEC protocol so as to ensure pretty goodstate transfer for small disorder strengths and indicates a threshold beyond which QEC does nothelp in improving the fidelity of state transfer.
I. INTRODUCTION
Quantum communication entails transmission of an ar-bitrary quantum state from one spatial location to an-other. Spin chains are a natural medium for quantumstate transfer over short distances, with the dynamics ofthe transfer being governed by the Hamiltonian describ-ing the spin-spin interactions along the chains. Startingwith the original proposal by Bose [1] for state transfervia a 1-d Heisenberg chain, several protocols have beendeveloped for perfect as well as pretty good quantum statetransfer via spin chains.Perfect state transfer protocols typically involve engi-neering the coupling strengths between the spins in sucha way as to ensure perfect fidelity between the state ofsender’s spin and that of the receiver’s spin [2–6]. Al-ternately, there have been proposals to use multiple spinchains in parallel, and apply appropriate encoding anddecoding operations at the sender and receiver’s spins soas to transmit the state perfectly [7–9]. Experimentally,perfect state transfer protocols have been implementedin various architectures including nuclear spins [10] andphotonic lattices using coupled waveguides [11, 12].Relaxing the constraint of perfect state transfer, pro-tocols for pretty good transfer aim to identify optimalschemes for transmitting information with high fidelityacross permanently coupled spin chains [13, 14]. Oneapproach is for example to encode the information as aGaussian wave packet in multiple spins at the sender’send [15, 16]. Moving away from ideal spin chains, quan-tum state transfer has also been studied over disorderedchains, both with random couplings and as well as ran-dom external fields [8, 17, 18].Here, we study the problem of pretty good state trans-fer from a quantum channel point of view. It is known [1]that state transfer over an ideal
XXX chain (also calledthe Heisenberg chain) can be realized as the action of anamplitude damping channel [19] on the encoded state.Naturally, this leads to the question of whether quan-tum error correction (QEC) can improve the fidelityof quantum state transfer. QEC-based protocols thatachieve pretty good transfer have been developed for noisy XX [20, 21] and Heisenberg spin chains [22].In our work we study the role of adaptive QEC inachieving pretty good transfer over a class of 1-d spinsystems which preserve the total spin. This includes boththe XX as well as the Heisenberg chains, and more gen-erally, the XXZ chain. We use an approximate
QEC(AQEC) code, which has been shown to achieve thesame level of fidelity as perfect QEC codes for certainnoise channels while making use of fewer physical re-sources [23–26]. Our protocol involves the use of multipleidentical spin chains in parallel, with the information en-coded in an entangled state across the chains. This isin contrast to the protocols in [20, 21] which use perfectQEC codes and encode into multiple spins on a singlechain. Using the worst-case fidelity between the statesof the sender and receiver’s spins as the figure of merit,we demonstrate that pretty good state transfer maybeachieved over a class of spin-preserving Hamiltonians us-ing an approximate code and a channel-adapted recoverymap.Finally, we present explicit results for the fidelity ofstate transfer obtained using our QEC scheme, for idealas well as disordered
XXX chains. The presence of dis-order in a 1-d spin chain is known to lead to the phe-nomenon of localization [27]. Here, we analyze the distri-bution of the transition amplitude for a disordered
XXX chain, with random coupling strengths which are drawnfrom a uniform distribution. We modify the QEC proto-col suitably so as to ensure pretty good transfer when thedisorder strength is small. As the disorder strength in-creases, our analysis points to a threshold beyond whichQEC does not help in improving the fidelity of statetransfer.The rest of the paper is organized as follows. Wediscuss the basic state transfer protocol over a generalclass of spin-preserving Hamiltonians and the underly-ing quantum channel description in Sec. II. We discussthe adaptive QEC protocol and the resulting fidelity inSec. III. We present results specific to the ideal
XXX chain in Sec. IV and discuss the disordered chain inSec. V. Finally, we summarize our conclusions in Sec. VI. a r X i v : . [ qu a n t - ph ] N ov II. PRELIMINARIES
We consider a general 1-d spin chain with nearestneighbour interactions described by the Hamiltonian, H = − (cid:88) k J k (cid:0) σ kx σ k +1 x + σ ky σ k +1 y (cid:1) − (cid:88) k ˜ J k σ kz σ k +1 z + (cid:88) k B k σ zk , (1)where, { J k } > { ˜ J k } > { B k } denote the magnetic field strengths at each site, and,( σ kx , σ ky , σ kz ) are the Pauli operators at the k th site. Thespin sites are numbered as j = 1 , , . . . , N . We assumethat the sender’s site is the s th spin and receiver’s site isthe r th spin.We denote the ground state of the spin as | (cid:105) = | . . . (cid:105) . Since we are interested in transmitting aqubit worth of information along the chain, we will workwithin the subspace spanned by the set of single parti-cle excited states | j (cid:105) , with | j (cid:105) denoting the state withthe j th spin alone flipped to | (cid:105) . The Hamiltonian inEq. (1) preserves the total number of excitations, that is, (cid:104) H , (cid:80) Ni =1 σ iz (cid:105) = 0 and hence the resulting dynamics isrestricted to the ( N + 1)-dimensional subspace spannedby the single particle excited states and the ground state.The sender encodes an arbitrary quantum state | ψ in (cid:105) = a | (cid:105) + b | (cid:105) at the s th site, with the coefficients a and b parameterized using a pair of angles ( θ, φ ) as a =cos( θ ), b = e − iφ sin( θ ). The initial state of the spin chain is thusgiven by, | Ψ(0) (cid:105) = a | (cid:105) + b | s (cid:105) , (2)where | s (cid:105) is the state of the spin chain with only the s th spin is flipped to | (cid:105) and all other spins set to | (cid:105) . Underthe action of the Hamiltonian H described in Eq. (1),after time t , the spin chain evolves to the state (here,and in what follows, we set (cid:126) = 1), | Ψ( t ) (cid:105) = e − i H t | Ψ(0) (cid:105) , = a | (cid:105) + b N (cid:88) j =1 (cid:104) j | e − i H t | s (cid:105)| j (cid:105) . Following [1], the state of the receiver’s spin at the r th site after time t , denoted as ρ out ( t ), is obtained by tracingout all the other spins from the state of the full spin chain ρ ( t ) = | Ψ( t ) (cid:105)(cid:104) Ψ( t ) | : ρ out ( t ) = tr , ,...,r − ,r +1 ,N − [ ρ ( t )]= (cid:2) | a | + | b | (cid:0) − | f Nr,s ( t ) | (cid:1)(cid:3) | (cid:105)(cid:104) | + ab ∗ ( f Ns,r ( t )) ∗ | (cid:105)(cid:104) | + ba ∗ f Nr,s ( t ) | (cid:105)(cid:104) | + | b | | f Nr,s ( t ) | | (cid:105)(cid:104) | , (3)where, f Nr,s ( t ) = (cid:104) r | e ( − i H t ) | s (cid:105) (4) is the transition amplitude , which gives the probabilityamplitude for the excitation to transition from the s th site to r th site. The function f Nr,s ( t ) satisfies, N (cid:88) r =1 | f Nr,s ( t ) | = 1 , ∀ s = 1 , , . . . , N. N (cid:88) k =1 f Nr,k ( t )( f Nk,s ( t )) ∗ = δ rs , ∀ k = 1 , , . . . , N. (5)where δ rs is the delta function with δ rs = 1 for r = s and δ rs = 0 for r (cid:54) = s .As shown in [1], we thus obtain the reduced state inEq. (3) at receiver’s end as the action of a quantum chan-nel on the input state. Specifically, ρ out ( t ) = E ( ρ in ) = (cid:88) k E k ρ in E † k , (6)where E and E are the Kraus operators that describethe action of the channel. It is easy to see that the oper-ators E , E have the following form when written in the {| (cid:105) , | (cid:105)} basis. E = (cid:18) f Nr,s ( t ) (cid:19) , E = (cid:32) (cid:113) − | f Nr,s ( t ) | (cid:33) . (7)The Kraus operators in Eq.(7) lead to a channel that hasthe same structure as the amplitude damping channel,but is more general since the parameter f Nr,s ( t ) charac-terizing the noise in the channel is complex.Recall that the standard amplitude damping channelis parameterized by a real noise parameter p and is de-scribed by a pair of Kraus operators, written in the {| (cid:105) , | (cid:105)} basis as [19], E AD0 = (cid:18) √ − p (cid:19) , E AD1 = (cid:18) √ p (cid:19) . (8)This is the quantum channel induced in the original statetransfer protocol in [1] where the Hamiltonian consideredis a Heisenberg chain in the presence of an external fieldof the form (cid:126)B = B ˆ z , that is,˜ H = − J (cid:88) (cid:104) i,j (cid:105) (cid:126)σ i · (cid:126)σ j − B (cid:88) i σ z . (9)By choosing the intensity of the (cid:126)B -field appropriately, itis possible to adjust the phase of the complex amplitude f Nr,s ( t ) to be a multiple of 2 π and hence replace f Nr,s ( t ) by | f Nr,s ( t ) | , thus obtaining the amplitude damping channeldescribed in Eq. (8) above.While much of the past work on state transfer has fo-cused on the Heisenberg Hamiltonian in Eq. (9), here, wewill focus on the more general Hamiltonian in Eq. (1). Westudy the problem of transmitting an arbitrary quantumstate from the s th site to the r th site of an N -spin chain.We quantify the performance of the protocol in terms ofthe fidelity between the final state ρ out ≡ E ( | ψ in (cid:105)(cid:104) ψ in | )and the input state | ψ in (cid:105) . Specifically, we use the worst-case fidelity, which is defined as [19], F ( E ) = min a,b (cid:104) ψ in | ρ out | ψ in (cid:105) , where the minimization is over all possible input states a | (cid:105) + b | (cid:105) . We say that pretty good state transfer isachieved when the worst-case fidelity F ( E ) ≥ − (cid:15) ,for some (cid:15) > | f Nr,s ( t ) | and Θ refer to the amplitude and phaserespectively, of the noise parameter f Nr,s ( t ) = e i Θ | f Nr,s ( t ) | of the general quantum channel in Eq. (7). For such achannel, the worst-case fidelity depends on both the am-plitude | f Nr,s ( t ) | as well as the phase Θ. However, follow-ing the original protocol in [1], if we choose the magneticfields { B k } so as to ensure that Θ is a multiple of 2 π , wecan show that, F ( E ) = | f Nr,s ( t ) | . (10)In what follows, we examine how the worst-case fidelitymay be improved using techniques from quantum errorcorrection. In particular, by obtaining a functional re-lationship between the worst-case fidelity and the tran-sition amplitude using an adaptive QEC procedure, weshow how the fidelity can be improved by an order in thenoise parameter. III. STATE TRANSFER PROTOCOL BASEDON ADAPTIVE QEC
Given a specific form of the spin-conserving Hamilto-nian in Eq. (1), it is possible to estimate | f Nr,s ( t ) | and Θfor a specific choice of sites s, r and t by making repeatedmeasurements on the spin chain [8]. Knowing Θ, we mayapply a phase gate of the form, U Θ = (cid:18) e − i Θ (cid:19) , (11)to change the encoding basis to {| (cid:105) , e − i Θ | (cid:105)} . In thisrotated basis, the channel in Eq. (7) is identical to theamplitude damping channel described in Eq. (8). At thelevel of the Hamiltonian, this is the same as choosingthe field strengths { B k } so as to make the phase Θ triv-ial. Indeed, by making an appropriate choice of mag-netic fields, it is always possible to transfom the spin-preserving Hamiltonian in Eq. (1) into an XXX interac-tion as in Eq. (9) (see [28]) and hence map the underlyingnoise channel to an amplitude-damping channel.One na¨ıve approach to improving the fidelity of statetransfer is to therefore first apply the U Θ -gate and thenuse any of the well known QEC protocols which correctfor amplitude damping noise [23–25, 29]. However, suchan approach fails in the presence of disorder. When weconsider a disordered 1-d spin chain wherein either thecouplings { J k , ˜ J k } or the fields { B k } in Eq. (1) maybe random, the underlying noise channel is stochastic. Thetwo real parameters | f Nr,s ( t ) | and Θ charcterizing the noisein the channel vary with each disorder realization, andhence an encoding procedure that relies on knowledge ofa specific realization of Θ is not useful. Moreover, imple-menting a phase gate as in Eq. (11)based on the disorder-averaged value of Θ does not help – such a phase gatewill no longer cancel out the arbitrary (random) phasein Eq. (7) and we do not obtain an amplitude dampingchannel in the rotated basis after the action of the phasegate.We would therefore like to tackle the problem of cor-recting for the more general noise channel in Eq. (7).Taking inspiration from the structural similarity to theamplitude damping channel, we propose a QEC protocolusing an approximate C , realized as the spanof the following pair of orthogonal states, | L (cid:105) = 1 √ | (cid:105) + | (cid:105) ) , | L (cid:105) = 1 √ | (cid:105) + | (cid:105) ) . (12)This code was shown to be approximately correctable foramplitude damping noise, both in terms of worst casefidelity [23] as well as entanglement fidelity [30]. Thecode is approximate in the sense it does not satisfy theconditions for perfect quantum error correction [19], forany single-qubit error.The recovery map we use is adapted to the given noisemap E and code C , and can be described in terms of theKraus operators of the noise and the projector P ontothe codespace, as follows, R ( . ) = (cid:88) i P E † i E ( P ) − / ( . ) E ( P ) − / E i P, (13)where the inverse of E ( P ) is taken on its support. Such arecovery map R has been shown to achieve worst-casefidelity close to that of the optimal recovery map forany given noise channel E [25]. In the specific case ofthe amplitude-damping channel and the 4-qubit code,the adaptive recovery map defined above was shown toachieve better worst-case fidelity than the recovery usedin [23]. FIG. 1: 4-qubit QEC on spin chainsThe quantum state transfer protocol with QEC is im-plemented using a set of 4 unmodulated, identical, spinchains. Fig. 1 depicts a schematic of our protocol. Theinitial, encoded state | ψ enc (cid:105) is now an entangled stateacross the four chains, involving only a single spin (the s th site)in each of the chains. | ψ enc (cid:105) = a | (cid:105) L + b | (cid:105) L . (14)Once the initial state is prepared, the four chains areallowed to evolve in an uncoupled fashion, according tothe Hamiltonian in Eq. (1). After time t , the state at thereceiver’s site is a joint state of the r th site of the fourchains, and is described by action of the map E ⊗ withthe time-dependent noise parameter f Nr,s ( t ). Thus, ρ err = E ⊗ ( ρ enc ) = (cid:88) i E (4) i ρ enc (cid:16) E (4) i (cid:17) † , where E (4) i are the Kraus operators of the 4-qubit noisechannel realized as four-fold tensor products of the op-erators E and E in Eq. (7). After evolving the chainsfor time t , the recovery map R (4) is applied at the re-ceiver’s site of the four spin chains. The final state at thereceiver’s end after the QEC protocol is obtained as, ρ rec = (cid:88) i,j R (4) j E (4) i ρ enc (cid:16) E (4) i (cid:17) † (cid:16) R (4) j (cid:17) † , with the Kraus operators R (4) i given by, R (4) i = P (cid:16) E (4) i (cid:17) † E ⊗ ( P ) − / , (15) where P ≡ | L (cid:105)(cid:104) L | + | L (cid:105)(cid:104) L | is the projector onto the4-qubit space described in Eq. (12). The fidelity of the4-chain quantum state transfer protocol is then given by, F (cid:16) R (4) ◦ E ⊗ , C (cid:17) ≡ min a,b (cid:104) ψ enc | ρ rec | ψ enc (cid:105) ., where the minimization is over all states in the codespace C . As before, pretty good transfer is achieved when theworst-case fidelity is high, that is, F (cid:0) R (4) ◦ E ⊗ , C (cid:1) ≥ − (cid:15) , for (cid:15) >
0. We now present a key result of the paper,namely a bound on the fidelity of state transfer usingthe adaptive QEC protocol, in terms of the transitionamplitude f Nr,s, ( t ). Theorem 1.
The fidelity of quantum state transfer fromsite s to site r under a spin-conserving Hamiltonian asin Eq. (1) , using the -qubit code C and adaptive recovery R (4) at time t , is given by, F (cid:16) R (4) ◦ E ⊗ , C (cid:17) ≈ − p O ( p ) , (16) where p = 1 − | f Nr,s ( t ) | . Proof. We first rewrite the Kraus operators given inEq. (7), as, E = | (cid:105)(cid:104) | + | f Nr,s ( t ) | e i Θ | (cid:105)(cid:104) | E = | (cid:105)(cid:104) | (cid:113) − | f Nr,s ( t ) | , where, | f Nr,s ( t ) | and Θ are the absolute value and phaseof the complex-valued transition amplitude f Nr,s ( t ). Thestate after the 4-qubit recovery map is then given by, ρ rec = (cid:16) R (4) ◦ E ⊗ (cid:17) ( ρ enc ) . The composite map (cid:0) R (4) ◦ E ⊗ (cid:1) comprising noise andrecovery has Kraus operators of the form, P (cid:16) E (4) j (cid:17) † E ⊗ ( P ) − / E (4) i P. (17) The key step in obtaining the desired fidelity is to showthat the Kraus operators of the composite map writ-ten above are independent of Θ. First, we write out E ⊗ ( P ) − / in the (standard) computational basis of the4-qubit space. E ⊗ ( P ) − / = (cid:88) i =1 G i | i (cid:105)(cid:104) i | + e − i Θ G | (cid:105)(cid:104) | + e i G | (cid:105)(cid:104) | + G ( | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ) , where {G i } are polynomial functions of the transi-tion amplitude | f Nr,s ( t ) | . The Θ-dependence in thispseudo-inverse operator occurs only in the span of {| (cid:105) , | (cid:105)} . Since E ⊗ ( P ) − / is sandwiched be-tween the Kraus operators of the 4-qubit channel andtheir adjoints, we also write down the Kraus operators { E (4) i } in the computational basis. Then, an explicitcomputation reveals that the Θ-dependence gets conju-gated out for each of the Kraus operators in Eq. (17). Werefer to Appendix A for the details of this calculation.Hence the final state after noise and recovery ρ rec can beexpressed as a linear sum of terms that are independentof Θ. Since the parameter Θ is effectively suppressed, thefidelity after using 4-qubit code and the universal recov-ery in Eq. (7), is purely a function of p = 1 − | f Nr,s ( t ) | .The fidelity corresponding to the initial state | ψ enc (cid:105) = a | L (cid:105) + b | L (cid:105) can thus be obtained as, F ( R (4) ◦ E ⊗ , C )= 1 − p (cid:0) ( | a | − | b | ) − (( ba ∗ ) + ( ab ∗ ) ) + 5 | a | | b | (cid:1) + O ( p ) , (18)where O ( p ) refers to terms of order p and higher.Parameterizing a and b as a = cos θ , b = e − iφ sin θ ,the fidelity attains its minimum value at { θ, φ } = { (2 n +1) π , (2 n +1) π } ( n = 1 , , . . . ), so that the worst-casefidelity over the 4-qubit code C is given by, F (cid:16) R (4) ◦ E ⊗ , C (cid:17) ≈ − p O ( p ) . (cid:4) Our result shows that using the adaptive recovery inconjunction with the approximate code leads to a fidelitythat is independent of the phase Θ of the complex noiseparameter f Nr,s ( t ). Thus, to optimize the fidelity of statetransfer between the s th and r th site of a chain of N spins evolving according to the Hamiltonian in Eq. (1),we simply need to find the time t at which | f Nr,s ( t ) | ismaximized. Recall that the worst-case fidelity withoutQEC (using the single chain protocol) is linear in theparameter p , as observed in Eq. (10). Thus we see an O ( p ) improvement in fidelity with QEC, as expected.Furthermore, our estimate of the worst-case fidelity im-plies that so long as the noise strength p is such that1 − (7 / p > − p , the adaptive QEC protocol achievesbetter fidelity than the single chain protocol withoutQEC. This constraints the noise strength p to satisfy0 < p < (4 / ≈ .
57. This in turn implies a thresholdfor the transition amplitude, namely, | f Nr,s ( t ) | > . IV. RESULTS FOR THE -D HEISENBERGCHAIN As a simple example to illustrate the performance ofthe adaptive QEC protocol, we now consider a specialcase of the Hamiltonian in Eq. (1), namely, an N -length,ideal Heisenberg chain, with J k = ˜ J k = J/ J >
0) and B k = 0, for all k . This is also often referred to as the XXX -chain in the literature. Setting J = 1 without lossof generality, we present numerical results on the fidelityof state transfer from the first ( s = 1) to the N th ( r = N )site. Single qubit ( no QEC ) - qubit ( after QEC ) - qubit ( after QEC ) Spin chain length N F m i n FIG. 2: Worst-case fidelity as a function of chain length N .Fig. 2 compares the performance of state transfer pro-tocols with and without QEC. In particular, it comparesthe performance of our 4-chain state transfer protocolwith the single-chain (no QEC) protocol [1] and the 5- chain protocol proposed in [22]. For each N , we plot thefidelity of state transfer from the 1 st site to the N th siteon a N -length spin chain, after a time t ∗ chosen such that | f NN, ( t ) | is maximum at t = t ∗ , for 0 < t < /J .From the plot we see that the QEC-based protocolsachieve pretty good state transfer over longer distancesthan the single chain protocol. Furthermore, using ap-proximate QEC it is possible to achieve as high as fi-delity as with the standard 5-qubit code, using fewer spinchains. Specifically, in the regime of small noise param-eter p , we can show that the worst-case fidelity obtainedusing the 5-qubit code is, F ≈ − p O ( p ) . (19)Correspondingly, a 5-chain protocol performs well oversingle chain protocol when 0 < p < (8 / ≈ . | f Nr,s ( t ) | > .
47, which is a higher threshold than thatrequired by our adaptive QEC protocol.For the ideal Heisenberg chain, it was recently shownthat [14], there always exists a time t at which | f N ,N ( t ) | > − (cid:15) if and only if the length of the chainis a power of 2, that is, N = 2 m . In other words, prettygood state transfer is always possible between the endsof a Heisenberg spin chain whose length N is of the form N = 2 m ( m > m sites. Specif-ically, we can achieve pretty good state transfer over achain of arbitrary length L , by stitching together smallerchains whose lengths are of the form N = 2 m . At everystage of the repeated QEC protocol, there are exactly 2 m interacting spins and the rest of the spin-spin-interactionsare turned off. Spin chain length N F m i n Single qubit ( Repeated, no QEC ) - qubit ( Repeated, with QEC ) FIG. 3: Worst-case fidelity using repeated QECFig. 3 shows an example of the resulting improvementin fidelity when the QEC protocol is repeated every 8sites. For comparison, we plot the worst-case fidelity ob-tained by stitching together a sequence of length-8 chains,without QEC. The repeated QEC protocol proceeds asfollows. We first implement our QEC protocol for an8-spin chain, evolving for time t ∗ at which | f , ( t ) | maxi-mizes. We repeat this procedure some k times, where k isthe largest integer such that 8 k < N and finally performQEC for the remaining N − k sites for the same waitingtime t ∗ . Such a repeated QEC protocol indeed enablespretty good transfer for much longer lengths, as seen inthe plot.More generally, if F ≈ − αp is the fidelity ofthe single-shot QEC protocol, repeating the procedure k times gives us a fidelity of F = 1 − ( p new ) , with, p new = (1 − (1 − αp ) k )where p new is the noise parameter obtained after repeat-ing QEC k times. V. QUANTUM STATE TRANSFER ON ADISORDERED HEISENBERG CHAIN
Moving away from an ideal spin chain with a fixed,uniform coupling between successive spins, we now studystate transfer over a disordered
XXX chain, where thespin-spin couplings are randomly drawn from some dis-tribution. It is well known that the presence of disorderin a 1-d spin chain leads to the phenomenon of localiza-tion [27] of information close to one end of the chain. It istherefore a challenging task to identify protocols whichachieve perfect or pretty good transfer over disorderedspin chains, overcoming the effects of localization.Past work on disordered chains has primarily focusedon the XX chain. Starting with a modulated chain thatadmits perfect state transfer, both random magnetic fieldand random couplings have been studied [18]. Alter-nately, an unmodulated chain with random couplings atall except the sender and receiver sites has also been stud-ied [17].When viewed in the quantum channel picture, the pres-ence of disorder becomes as an additional source of noise.The role of QEC in overcoming the effects of disorder hasbeen studied both for the XX [20] as well as the Heisen-berg chains [22]. The QEC protocol for a noisy XX chainwith random couplings involves encoding into multiplespins on a single chain using modified CSS codes [20].The QEC protocol in [22] encodes into multiple iden-tical, uncoupled chains using the standard 5-qubit code,while also requiring access to multiple spins at the senderand receiver ends of each of the chains. Furthermore, theprotocol based on the 5-qubit code involves choosing anencoding based on the phase Θ of the transition ampli-tude (as explained in Sec. III), which in turn is specific tothe disorder realization. This makes the QEC procedurehard to implement in a practical sense.Here, we show how the channel-adapted QEC proce-dure described in Sec. III can be used to achieve prettygood state transfer over an XXX chain with randomcouplings. As before, we quantify the performance of the state transfer protocol in terms of the fidelity between theinitial and final states. When the underlying quantumchannel is stochastic, as in the case of a disordered chain,we use the disorder-averaged worst-case fidelity (cid:104) F (cid:105) δ ,to characterize the performance of the state transfer pro-tocol. We say that pretty good state transfer is achievedby a certain choice of code C and recovery R when thecorresponding disorder-averaged fidelity (cid:104) F (cid:105) δ ≥ − (cid:15) ,for some (cid:15) > J k = J (1 + ∆ k ), where ∆ k are independent, identi-cally distributed random variables drawn from a uniformdistribution between [ − δ, δ ] and J is the mean value ofthe coupling strength, which we may set to 1, withoutloss of generality. Note that such a Hamiltonian con-serves the total spin and hence falls within the universal-ity class discussed in Sec. II.Consider a state transfer protocol, where the senderwishes to transmit the state | ψ in (cid:105) = a | (cid:105) + b | (cid:105) from the s th site to the r th site via the natural dynamics of thechain. As before, the final state at the receiver’s site,tracing out the other spins can be realized as the actionof a quantum channel E , ρ out = E ( ρ in ) = (cid:88) k E k ρ in E † k , with the same Kraus operators { E , E } as in Eq. (7).The key difference however is in the nature of the noiseparameter p ≡ − | f Nr,s ( t, { ∆ k } ) | : in the case of the disordered chain, the transition amplitude f Nr,s ( t, { ∆ k } )between site s and r for a chain of length N allowed toevolve for a time t , is a random variable whose value de-pends on the specific realization of the disorder variables { ∆ k } . The distribution of f Nr,s ( t, { ∆ k } ) for given set of r, s, N, t values depends on the distribution over whichthe disorder variables { ∆ k } are sampled. To illustrateour point, we specifically consider below the case wherethe coupling strengths { ∆ k } are independently sampledfrom a uniform distribution. A. Transition amplitude in the presence of disorder
The Heisenberg Hamiltonian H with static disorder inthe coupling strengths, has the form, H dis = − (cid:88) k J (1 + ∆ k )2 ( σ kx σ k +1 x + σ ky σ k +1 y + σ kz σ k +1 z ) . (20)Here, the effect of disorder is introduced via the i.i.d.random variables { ∆ i } which take values over a uniformdistribution between [ − δ, δ ]. The quantity δ is called thedisorder strength, and J is the mean value of the couplingcoefficient. We may view the disordered Hamiltonian asa sum of the form H dis = H o + H δ , where H o denotes theideal XXX
Hamiltonian studied in the previous sectionand H δ is given by, H δ = − J (cid:88) k ∆ k −→ σ k · −−−→ σ k +1 . H δ captures the effect of disorder in the spin chain andcan be treated as a perturbation of the Hamiltonian H .Since [ H , H δ ] (cid:54) = 0, the transition amplitude maybe eval-uated using the so-called time-ordered expansion, alsoreferred to as the Dyson-series [31].Specifically, the transition amplitude between the r th and s th site for the disordered Hamiltonian H dis inEq. (20) is given by (setting (cid:126) = 1), f Nr,s ( t, { ∆ k } )= (cid:104) r | e − i ( H o + H δ ) t | s (cid:105) = (cid:104) r | e − i H o t T (cid:20) exp (cid:18) − i (cid:90) t e i H o t (cid:48) H δ e − i H o t (cid:48) dt (cid:48) (cid:19)(cid:21) | s (cid:105) = f Nr,s ( t ) − i N (cid:88) k =1 f Nr,k ( t ) (cid:90) t (cid:104) k | e i H o t (cid:48) H δ e − i H o t (cid:48) | s (cid:105) dt (cid:48) + O ( H δ ) , where T is the time-ordering operator which has beenexpanded to first order in the perturbation in the finalequation. As before, f Nr,k ( t ) denotes the transition ampli-tude between the r th and k th sites in the case of an idealchain of length N , without disorder.Thus, using the time-ordered expansion, the transitionamplitude in the presence of disorder can be evaluatedas a perturbation around the zero-disorder value f Nr,s ( t ),of the form, f Nr,s ( t, { ∆ k } ) = f Nr,s ( t )+ N − (cid:88) i =1 c Ni ( t )∆ i + N − (cid:88) i,j =1 d Nij ∆ i ∆ j + . . . . (21)The explicit forms of the complex coefficients c Ni ( t ) aregiven in Eq. (B6) in Appendix B. A similar approachwas used in [18] to study deviations from perfect statetransfer due to the presence of disorder in an XX chain.Using the form of the transition amplitude stated inEq. (21), we obtain the distribution of real part of thetransition amplitude x ≡ Re [ f Nr,s ( t, { ∆ k } )] , up to firstorder in the perturbation H δ , as, P δ,N,t ( x ) ∝ N − (cid:88) j =1 ( − u j ( q j ) N − Sign[ q j ] , (22)where u j ∈ [0 ,
1] and the
Sign function is defined asSign( x − a ) = − , x < a, , x = a, , x > a. The functions q j (cid:0) x, Re [ f Nr,s ( t )] , { Re [ c Ni ( t )] } (cid:1) are linearcombinations of the form, q j ≡ x − Re [ f Nr,s ( t )] + δ N − (cid:88) i =1 ( − r ji Re [ c Ni ( t )] , (23) - - Re [ f ( ) ( t * , { Δ k })] P ( R e [ f ( , ) ( t * , { Δ k } )]) FIG. 4: Distribution of Re [ f , ( t ∗ , { ∆ k } )] for differentdisorder realizations, drawn from a uniform distributionwith disorder strength δ = 0 . r ji ∈ [0 , ∀ i = 1 , . . . , N − Re [ c Ni ( t )] denotethe real part of the coefficients in Eq. 21. Since there are N − N , thesum over i ranges from 1 to N −
1. There are 2 N − dis-tinct linear combinations of the form q j , correspondingto the 2 N − distinct ( N − r j , so that the sum over j runs from 1 to 2 N − .The form of the distribution is identical for the imaginarypart Im [ f Nr,s ( t, { ∆ k } )], with the real parts of { c Ni ( t ) } and f Nr,s ( t ) replaced by their imaginary parts. We refer toEqs. (B9), (B11) in Appendix B for a detailed descrip-tion of the distributions of the real and imaginary partsof the disordered transition amplitude.The key salient feature we observe from calculating thedistribution functions above is that the limiting distribu-tion in the case of no disorder ( δ → f Nr,s ( t ). Furthermore, in Ap-pendix B we also explicitly evaluate the mean and vari-ance of f Nr,s ( t, { ∆ k } ) and show that the mean is equalto the zero-disorder value of f Nr,s ( t ), up to O ( δ ) (seeEq. B15). The variance goes as O ( δ ), as shown inEq. (B17), making it vanishingly small in the limit ofsmall δ . This observation leads us to propose a modifiedQEC protocol for state transfer over disordered XXX chains, using an adaptive recovery R avg based on the disorder-averaged transition amplitude (cid:104) f Nr,s ( t, { ∆ k } ) (cid:105) δ .The analysis presented thus far holds for any pair ofsites ( s, r ) on a spin chain of length N . As an exam-ple, we consider the specific case of an 8-length chain,with s = 1 and r = 8. We plot the distribution ofthe real part of the transition amplitude at some fixedtime t ∗ , for disorder strengths δ = 0 .
001 and δ = 1, inFigs. 4 and 5 respectively. We see that when the disor-der strength is small enough, the transition amplitude isindeed distributed like a delta function peaked aroundthe zero-disorder value. For large values of δ , the distri-bution spreads out quite a bit and its mean also shiftscloser to zero, giving rise to a very small transition am- - Re [ f ( ) ( t * , { Δ k })] P ( R e [ f ( , ) ( t * , { Δ k } )]) FIG. 5: Distribution of Re [ f , ( t ∗ , { ∆ k } )] for differentdisorder realizations, with disorder strength δ = 1.plitude. The corresponding figures for Im [ f , ( t ∗ , { ∆ k } )]are presented in Fig. 8. B. Adaptive QEC for -d disordered chain To summarize, the quantum channel for state transferin the presence of disorder has the same structure as thatof the ideal chain, but with a stochastic noise parameter p ≡ − | f Nr,s ( t, { ∆ k } ) | , since the transition amplitude f Nr,s ( t, { ∆ k } ) is now a random variable whose value de-pends on the random couplings { ∆ k } . However, as dis-cussed in Sec. V A, for small enough disorder strengths, f Nr,s ( t, { ∆ k } ) is peaked sharply around its mean value,and we may consider the disorder-averaged amplitude (cid:104) f Nr,s ( t, { ∆ k } ) (cid:105) δ as a good estimate of the noise.We therefore propose an adaptive QEC procedure fora disordered XXX chain involving the 4-qubit code inEq. (12) and a recovery map R avg with the same struc-ture as that used in the case of the ideal chain, de-scribed in Eq. (15). However, unlike the ideal case,the value of the channel parameter used in the recov-ery is different from the one in actual noise channel :the recovery map uses the disorder-averaged amplitude (cid:104) f Nr,s ( t, { ∆ k } ) (cid:105) δ , and is therefore independent of the spe-cific disorder realization, whereas the noise channel hasthe parameter f Nr,s ( t, { ∆ k } ) which changes with every re-alization.To illustrate the performance of this modified recov-ery map, we present numerical results for quantum statetransfer from the first site ( s = 1) to the 8 th site( r = 8)on an 8-spin chain. Fig. 6 shows the disorder-averaged worst-case fidelity (cid:104) F (cid:105) δ obtained using the4-qubit code and the adaptive recovery R avg , for an 8-spin chain. For disorder strengths δ ≤ .
01, our adap-tive QEC protocol achieves pretty good transfer, withfidelity-loss (cid:15) < .
2. Beyond δ ≥ .
06, we notice that (cid:104) F (cid:105) δ < . ● ● ● ● ● ● ●■ ■ ■ ■ ■ ■ ■◆ ◆ ◆ ◆ ◆ ◆ ◆▲ ▲ ▲ ▲ ▲ ▲ ▲● δ = ■ δ = ◆ δ = ▲ δ = Spin chain length N < F m i n > δ FIG. 6: Disorder-averaged worst-case fidelity (cid:104) F (cid:105) δ obtained using the adaptive recovery R avg .This is further borne out by our detailed analysis of thedistribution of the transition amplitude in the presenceof disorder (see Appendix B). In particular, our expres-sions for the mean and standard deviation of the transi-tion amplitude indicate that until δ ≤ .
01, the disorder-averaged value (cid:104) f Nr,s ( t, { ∆ k } ) (cid:105) δ is close to the value of thetransition amplitude in the ideal (zero-disorder) case, andthe standard deviation is insignificant compared to themean. However, as the disorder strength increases fur-ther, the disorder-averaged value (cid:104) f Nr,s ( t, { ∆ k } ) (cid:105) δ startsdropping and the standard deviation becomes compara-ble to the average value. Thus the effective noise pa-rameter of the underlying quantum channel becomes toostrong for the QEC procedure to be effective.The fact that δ = 0 .
06 is a threshold of sortscan be seen more directly by studying the variationof the disorder-averaged transition amplitude with dis-order strength. Previous studies on localization indisordered chains have used such a quantity, namely (cid:104)| f Nn, ( t, { ∆ k } ) | (cid:105) δ , as an indicator of the extent of lo-calization [17, 18]. ●●●●●●●● ■■■■■■■■ ◆◆◆◆◆◆◆◆ ▲▲▲▲▲▲▲▲ ▼▼▼▼▼▼▼▼● δ = ■ δ = ◆ δ = ▲ δ = ▼ δ = Site n on the spin chain of length N = < | f ( n , ) ( t * , { Δ k } ) > δ FIG. 7: (cid:104)| f n, ( t ) | (cid:105) δ for an 8-spin chain as a function ofthe site n .In Fig. 7, we plot the disorder-averaged transition am-plitude (cid:104)| f Nn, ( t, { ∆ k } ) | (cid:105) δ for a fixed time t ∗ and differentdisorder strengths δ , as a function of the receiver site n ,for the Heisenberg chain in Eq. (20). Empirically, we seethat this plot follows an exponential distribution. Thecurves take the form e − ( αn + β ) / Loc , where α, β are func-tions of disorder strength δ and Loc is the localizationlength, i.e. the length at which (cid:104)| f Nn, ( t, { ∆ k } ) | (cid:105) δ falls to(1 /e ) of its maximum value. We see that with increase indisorder strength δ , the localization effects become morepronounced.Specifically, when the disorder strength crosses δ =0 .
06, the square of the transition amplitude between theends of the 8-spin chain falls below 0 .
43 on average. How-ever, we know from the fidelity estimate in Theorem 1that the adaptive QEC protocol improves fidelity if onlyif | f NN, ( t ) | > .
43. Thus, for end to end state transferon an 8-length disordered Heisenberg chain, δ = 0 .
06 isindeed a threshold beyond which the adaptive QEC pro-tocol cannot help in improving fidelity. Since our analysisof the distribution of the transition amplitude presentedin Sec. V A as well as the fidelity expression in Theo-rem 1 hold for any s, r, N we can always identify such athreshold for a specific set of values.
VI. CONCLUSIONS
We develop a pretty good state transfer protocol basedon adaptive quantum error correction (QEC), for a uni-versal class of Hamiltonians which preserve the total spinexcitations on a linear spin chain. Based on the structureof the underlying quantum channel, we choose an approx-imate code and near-optimal, adaptive recovery map, tosolve for the fidelity of state transfer explicitly. For thespecific case of the ideal Heisenberg chain, our protocolperforms as efficiently as perfect-QEC-based protocols.Using repeated QEC on the chain, we are able to achievehigh enough fidelity over longer distances for an idealspin chain.In the case of disordered spin chains the underlyingquantum channel is stochastic. For the case of a disor-dered 1-d Heisenberg chain, we study the distribution ofthe transition amplitude, which in turn is directly relatedto the stochastic noise parameter of the noise channel.By suitably adapting the recovery procedure, we demon-strate pretty good transfer on average, for low disorderstrengths.It is an interesting question as to whether such channel-adapted QEC techniques maybe used to achieve prettygood state transfer for other universal classes, such as thetransverse-field Ising model and the
XY Z -chain. It isalso an open problem to obtain an efficient circuit imple-mentation of the adaptive recovery map discussed here.
VII. ACKNOWLEDGEMENTS
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Appendix A: Effect of noise channel E on -qubit code We note the following structure for the Kraus operators of the 4-qubit channel, by expanding them in the 4-qubitcomputational basis. First, we note that the only Kraus operator diagonal in the computational basis is E ⊗ , withdiagonal entry e ij Θ | f Nr,s ( t ) | j , corresponding to those basis vectors with j E in three of the four qubits) is of the form, E ⊗ E ⊗ = (1 − | f Nr,s ( t ) | ) / | (cid:105)(cid:104) | + e i Θ | f Nr,s ( t ) | (1 − | f Nr,s ( t ) | ) / | (cid:105)(cid:104) | . (A1)The remaining three-qubit errors are of the same form, with the strings { , } replaced by their permutations.Similarly, an operator which has E errors on two of the qubits is a linear combination of the form, E ⊗ ⊗ E ⊗ = (1 − | f Nr,s ( t ) | ) | (cid:105)(cid:104) | + e i Θ | f Nr,s ( t ) | (1 − | f Nr,s ( t ) | ) | (cid:105)(cid:104) | + e i Θ | f Nr,s ( t ) | (1 − | f Nr,s ( t ) | ) ( | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ) . (A2)Other two-qubit error operators are realized by replacing the strings { , , , } with permutationsthereof. A single-qubit error operator, with E error on only one of the qubits has the form, E ⊗ ⊗ E = (cid:113) − | f Nr,s ( t ) | | (cid:105)(cid:104) | + e i Θ | f Nr,s ( t ) | (cid:113) − | f Nr,s ( t ) | ( | (cid:105)(cid:104) | + | (cid:105)(cid:104) | + | (cid:105)(cid:104) | )+ e i Θ | f Nr,s ( t ) | (cid:113) − | f Nr,s ( t ) | ( | (cid:105)(cid:104) | + | (cid:105)(cid:104) | + | (cid:105)(cid:104) | )+ e i Θ | f Nr,s ( t ) | (cid:113) − | f Nr,s ( t ) | | (cid:105)(cid:104) | . (A3)Finally, the four-qubit error operator E ⊗ is of the form, E ⊗ = (1 − | f Nr,s ( t ) | ) | (cid:105)(cid:104) | . (A4)We next explicitly write out the operator E ⊗ ( P ) in the computational basis of the 4-qubit space. E ⊗ ( P ) = Q e − i Θ Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q e i Θ Q Q , {Q i } denoting polynomial functions of the transition amplitude | f Nr,s ( t ) | . In terms of the rank-1 projectors ontothe computational basis states, we may write E ⊗ ( P ) as, E ⊗ ( P ) = (cid:88) i =1 Q i | i (cid:105)(cid:104) i | + e − i Θ Q | (cid:105)(cid:104) | + e i Q | (cid:105)(cid:104) | + Q ( | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ) , (A5) wherein | i (cid:105) ∈ {| (cid:105) , . . . , | (cid:105) , . . . , | (cid:105)} denote the computational basis states of the 4-qubit space.Similarly, we can also express the pseudo-inverse E ⊗ ( P ) − / in the 4-qubit computational basis, as follows: E ⊗ ( P ) − / = G e − i Θ G G G G G G G G G G G G G G G G G e i Θ G G , (A6) with {G i } denoting a set of polynomials in | f Nr,s ( t ) | . In terms of the rank-1 projectors onto the computational basisstates, we have, E ⊗ ( P ) − / = (cid:88) i =1 G i | i (cid:105)(cid:104) i | + e − i Θ G | (cid:105)(cid:104) | + e i G | (cid:105)(cid:104) | + G ( | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ) . (A7) Upon sandwiching the operator in Eq. (A7) between the different error operators of the four-qubit noise channel (asdescribed in Eqs. (A1), (A2), (A3), (A4)) and their adjoints, it is easy to see that the phases cancel out everywhere.In other words, the Kraus operators of the composite channel comprising noise and recovery are all independent ofthe phase Θ of the transition amplitude.
Appendix B: Distribution of the transition amplitude for a disordered
XXX chain
Here we derive the distribution of the transition amplitude f Nr,s ( t, { ∆ k } ) for the disordered XXX chain describedin Eq. (20), as a function of time t and disorder strength δ . Recall that the transition amplitude between the r th and s th site for the disordered Hamiltonian H dis is given by, f Nr,s ( t, { ∆ k } ) = (cid:104) r | e − i ( H o + H δ ) t | s (cid:105) = (cid:104) r | e − i H o t T (cid:20) exp (cid:18) − i (cid:90) t e i H o t (cid:48) H δ e − i H o t (cid:48) dt (cid:48) (cid:19)(cid:21) | s (cid:105) , (B1)where T denotes the time-ordering operator. We first expand the time-ordered perturbation series in Eq. (B1) asfollows, f Nr,s ( t, { ∆ k } ) = N (cid:88) k =1 (cid:104) r | e − i H o t | k (cid:105)(cid:104) k |T (cid:104) e ( − i (cid:82) t e i H ot (cid:48) H δ e − i H ot (cid:48) dt (cid:48) ) (cid:105) | s (cid:105) = N (cid:88) k =1 f Nr,k ( t ) (cid:104) k | I − iO ( H δ ) + i O ( H δ ) + . . . | s (cid:105) (B2)2where, f Nr,k ( t ) = (cid:104) r | e − i H o t | k (cid:105) is the transition amplitude in the absence of disorder. Expanding the first order term( O ( H δ )) as a time-ordered form, we have, (cid:104) k | O ( H δ ) | s (cid:105) = (cid:90) t (cid:104) k | e i H o t (cid:48) H δ e − i H o t (cid:48) | s (cid:105) dt (cid:48) = N (cid:88) l,m =1 (cid:90) t (cid:104) k | e i H o t (cid:48) | l (cid:105)(cid:104) l |H δ | m (cid:105)(cid:104) m | e − i H o t (cid:48) | s (cid:105) dt (cid:48) (B3)where, (cid:104) l |H δ | m (cid:105) = J (cid:32) N − (cid:88) i =1 ( u li ∆ i ) δ lm − l δ m ( l +1) − l − δ m ( l − (cid:33) , (B4)with the coefficients u li ∈ {± } . For example, H δ for a 4-qubit spin chain is a tridiagonal matrix of the form, H δ = J − ∆ − ∆ + ∆ − − − ∆ + ∆ + ∆ − − ∆ + ∆ − ∆ − − ∆ − ∆ − ∆ . Substituting the form of H δ in Eq. (B4) to the first order term in Eq. (B2), and setting J = 1 throughout, we get, f Nr,s ( t, { ∆ k } ) = f Nr,s ( t ) − i (cid:90) t N (cid:88) l,k =1 f Nr,k ( t )( f Nk,l ( t (cid:48) )) ∗ f Nl,s ( t (cid:48) )( N − (cid:88) i =1 u li ∆ i ) dt (cid:48) − i (cid:90) t N − (cid:88) l =1 N (cid:88) k =1 f Nr,k ( t )( f Nk,l ( t (cid:48) )) ∗ f Nl +1 ,s ( t (cid:48) )( − l ) dt (cid:48) − i (cid:90) t N − (cid:88) l =1 N (cid:88) k =1 f Nr,k ( t )( f Nk,l +1 ( t (cid:48) )) ∗ f Nl,s ( t (cid:48) )( − l ) dt (cid:48) . Thus, up to first order in perturbation, f Nr,s ( t, { ∆ k } ) is simply a linear combination of the random variables { ∆ k } , ofthe form, f Nr,s ( t, { ∆ k } ) = f Nr,s ( t ) + N − (cid:88) i =1 c Ni ( t )∆ i , (B5)where { c Ni ( t ) } are complex coefficients given by, c Ni ( t ) = − i N (cid:88) k =1 f Nr,k ( t ) (cid:34)(cid:90) t N (cid:88) l =1 u li ( f Nk,l ( t (cid:48) )) ∗ f Nl,s ( t (cid:48) ) dt (cid:48) − (cid:90) t ( f Nk,i ( t (cid:48) )) ∗ f Ni +1 ,s ( t (cid:48) ) dt (cid:48) − (cid:90) t ( f Nk,i +1 ( t (cid:48) )) ∗ f Ni,s ( t (cid:48) ) dt (cid:48) (cid:35) . (B6)We first note that in the limit of large N , the distribution of f Nr,s ( t ) tends towards a normal distribution. This is adirect consequence of the central limit theorem, since { ∆ i } are i.i.d random variables. In what follows, we will obtainthe exact form of the distribution of f Nr,s ( t, { ∆ k } ), specifically, the real and imaginary parts of f Nr,s ( t, { ∆ k } ) in termsof N, t and δ .Since the { ∆ i } are randomly drawn from a uniform distribution between [ − δ, δ ], the joint probability density P (∆ , ∆ , . . . , ∆ N ) is given by, P ( ∆ , ∆ , . . . , ∆ N − ) = (cid:26) δ ) N − , − δ ≤ ∆ i ≤ δ, ∀ i = 1 , , . . . , N − . , otherwise . (B7)Let x ≡ Re [ f Nr,s ( t, { ∆ k } )] and y ≡ Im [ f Nr,s ( t, { ∆ k } )] denote the real and imaginary parts of the transition amplitude inEq. (B5). Then, we may obtain the distribution of x and y as follows: P δ,t,N ( x ) = (cid:90) δ ∆ = − δ . . . (cid:90) δ ∆ N − = − δ (cid:32) N − (cid:89) i =1 d ∆ i (cid:33) P (∆ , ∆ , . . . , ∆ N − ) δ (cid:32) x − ( Re [ f Nr,s ( t )] + N − (cid:88) i =1 Re [ c Ni ( t )]∆ i ) (cid:33) , P δ,t,N ( y ) = (cid:90) δ ∆ = − δ . . . (cid:90) δ ∆ N − = − δ (cid:32) N − (cid:89) i =1 d ∆ i (cid:33) P (∆ , ∆ , . . . , ∆ N − ) δ (cid:32) y − ( Im [ f Nr,s ( t )] + N − (cid:88) i =1 Im [ c Ni ( t )]∆ i ) (cid:33) . { ∆ k } variables, weget, P δ,t,N ( x ) = 1 √ π (2 δ ) N − (cid:90) δ ∆ = − δ . . . (cid:90) δ ∆ N − = − δ (cid:90) ∞ k = −∞ N − (cid:89) i =1 d ∆ i dk exp (cid:32) − ik (cid:32) x − (cid:34) Re [ f Nr,s ( t )] + N − (cid:88) i =1 Re [ c Ni ( t )]∆ i (cid:35)(cid:33)(cid:33) = 1 √ π (2 δ ) N − (cid:90) ∞ k = −∞ dk exp (cid:0) − ik ( x − Re [ f Nr,s ( t )] ) (cid:1) N − (cid:89) i =1 (cid:0) kδ Re [ c Ni ( t )] (cid:1) k Re [ c Ni ( t )] (B8)= 1 √ π (2 δ ) N − (cid:90) ∞ k = −∞ dk exp (cid:0) − ik ( x − Re [ f Nr,s ( t )] ) (cid:1) N − (cid:89) i =1 e i ( kδ Re [ c Ni ( t )] ) − e − i ( kδ Re [ c Ni ( t )] ) ik Re [ c Ni ( t )]= 1 √ π (2 δ ) N − (cid:90) ∞ k = −∞ dk exp (cid:0) − ik ( x − Re [ f Nr,s ( t )] ) (cid:1) (cid:80) N − j =1 ( − α j e i ( k δ (cid:80) N − i =1 ( − rji Re [ c Ni ( t )]) ( ik ) N − (cid:81) N − i =1 Re [ c Ni ( t )] , where, α j , r ji ∈ [0 , , ∀ i, j . Simplifying further, we get, P δ,t,N ( x ) = 1 √ π (2 δ ) N − (cid:81) N − i =1 Re [ c Ni ( t )] (cid:90) ∞ k = −∞ dk (cid:80) N − j =1 ( − α j exp (cid:16) − ik ( x − Re [ f Nr,s ( t )] + δ (cid:80) N − i =1 ( − r ji Re [ c Ni ( t )]) (cid:17) ( ik ) N − = (cid:18) δ ) N − (cid:19) (cid:32) (cid:81) N − i =1 Re [ c Ni ( t )] (cid:33) N − (cid:88) j =1 ( − u j ( q j ) N − Sign[ q j ] , (B9)where u j ∈ [0 , q j ( x, Re [ f Nr,s ( t )] , { Re [ c Ni ( t )] } ) are linear combinations of the form, q j ≡ x − Re [ f Nr,s ( t )] + δ N − (cid:88) i =1 ( − r ji Re [ c Ni ( t )] , r ji ∈ [0 , , ∀ i = 1 , . . . , N − . (B10)We may evaluate the distribution of the imaginary part of the transition amplitude in a similar fashion, to get, P δ,t,N ( y ) = (cid:18) δ ) N − (cid:19) (cid:32) (cid:81) N − i =1 Im [ c Ni ( t )] (cid:33) N − (cid:88) i =1 ( − u j (˜ q j ) N − Sign[˜ q j ] , (B11)where the ˜ q j ( x, Im [ f Nr,s ( t )] , { Im [ c Ni ( t )] } ) are linear combinations of the form,˜ q j ≡ y − Im [ f Nr,s ( t )] + δ N − (cid:88) i =1 ( − r ji Im [ c Ni ( t )] , r ji ∈ [0 , , ∀ i = 1 , . . . , N − . (B12)We see from Eq. (B8) that the limiting distribution in the case of no disorder ( δ → Re [ f Nr,s ( t )]:lim δ → P δ,t,N ( x ) = 1 √ π (cid:90) ∞ k = −∞ dk exp (cid:0) − ik ( x − Re [ f Nr,s ( t )] ) (cid:1) = δ (cid:0) x − Re [ f Nr,s ( t )] (cid:1) . (B13)4 - - Im [ f ( ) ( t * , { Δ k })] P ( I m [ f ( , ) ( t * , { Δ k } )]) - Im [ f ( ) ( t * , { Δ k })] P ( I m [ f ( , ) ( t * , { Δ k } )]) FIG. 8: Distribution of Im [ f , ( t ∗ , { ∆ k } )] over different disorder realizations drawn from a uniform distribution withdisorder strengths δ = 0 .
001 and δ = 1, respectively.Finally, we compute the disorder-averaged value of the transition amplitude upto O ( H δ ). We first modify theexpression in Eq. (B5) to include the second-order perturbation terms: f Nr,s ( t, { ∆ k } ) = f Nr,s ( t ) + N − (cid:88) i =1 c Ni ( t )∆ i + N − (cid:88) i,j =1 d Nij ∆ i ∆ j + . . . , (B14)where { d Nij } are complex coefficients which are convolutions of the zero-disorder transition amplitude, similar to { c Ni ( t ) } . Next, using the fact that the random couplings { ∆ i } are drawn from a uniform distribution, we obtain, (cid:104) f Nr,s ( t, { ∆ k } ) (cid:105) δ = 1(2 δ ) N − (cid:90) δ − δ f Nr,s ( t ) + N − (cid:88) i =1 c Ni ( t )∆ i + N − (cid:88) l,m =1 d Nlm ( t )∆ l ∆ m + . . . N − (cid:89) i =1 d ∆ i = f Nr,s ( t ) + δ (cid:88) i d Nii ( t ) + O ( δ ) . (B15)The second moment of f Nr,s ( t, { ∆ k } ), (cid:10) ( f Nr,s ( t, { ∆ k } )) (cid:11) δ = 1(2 δ ) N − (cid:90) δ − δ f Nr,s ( t ) + N − (cid:88) i =1 c Ni ( t )∆ i + N − (cid:88) l,m =1 d Nlm ( t )∆ l ∆ m ) + . . . N − (cid:89) i =1 d ∆ i = ( f Nr,s ( t )) + δ f Nr,s ( t ) N − (cid:88) l =1 d Nll ( t ) + N − (cid:88) j =1 ( c Nj ( t )) + δ N − (cid:88) l =1 ( d Nll ( t )) + δ N − (cid:88) l (cid:54) = m =1 ( d Nlm ( t )) + O ( δ ) . (B16)We can now calculate the variance from Eq B15 and Eq B16 as follows:Var[ f Nr,s ( t, { ∆ k } )] = (cid:104) ( f Nr,s ( t, { ∆ k } )) (cid:105) δ − (cid:104) f Nr,s ( t, { ∆ k } ) (cid:105) δ = δ N − (cid:88) j =1 ( c Nj ( t )) + δ N − (cid:88) l =1 ( d Nll ( t )) + 19 N − (cid:88) l (cid:54) = m =1 ( d Nlm ( t )) − (cid:32) N − (cid:88) l =1 d Nll ( t ) (cid:33) + O ( δ ) . (B17)To summarize, from Eq. (B15) we see that as δ → (cid:104) f Nr,s ( t, { ∆ k } ) (cid:105) δ approaches the zero-disorder value f Nr,s ( t ).As expected, the variance given in Eq. (B17) vanishes in this limit. However as the disorder strength δ increases,5 (cid:104) f Nr,s ( t, { ∆ k } ) (cid:105) δ deviates from the no-disorder case, and the variance also starts growing since terms of O ( δ2