Nearly optimal time-independent reversal of a spin chain

We propose a time-independent Hamiltonian protocol for the reversal of qubit ordering in a chain of $N$ spins. Our protocol has an easily implementable nearest-neighbor, transverse-field Ising model Hamiltonian with time-independent, non-uniform couplings. Under appropriate normalization, we implement this state reversal three times faster than a naive approach using SWAP gates, in time comparable to a protocol of Raussendorf [Phys. Rev. A 72, 052301 (2005)] that requires dynamical control. We also prove lower bounds on state reversal by using results on the entanglement capacity of Hamiltonians and show that we are within a factor $1.502(1+1/N)$ of the shortest time possible. Our lower bound holds for all nearest-neighbor qubit protocols with arbitrary finite ancilla spaces and local operations and classical communication. Finally, we extend our protocol to an infinite family of nearest-neighbor, time-independent Hamiltonian protocols for state reversal. This includes chains with nearly uniform coupling that may be especially feasible for experimental implementation.

Quantum information transfer is a fundamental operation in quantum physics, and fast, accurate protocols for transferring quantum states across a physical system are likely to play a key role in the design of quantum computers and networks [2,3].For example, quantum information transfer can be used to establish long-range entanglement and is also useful for qubit routing in quantum architectures with limited connectivity [4,5].Extensive work has studied the implementation of various information transfer protocols, often via Hamiltonian dynamics on spin chains [6].
Information transfer in Hamiltonian systems is governed by the spread of entanglement and has close links to Lieb-Robinson bounds [7], entanglement area laws [8], and algorithms for quantum simulation [9].Fundamental limits to the rate of entanglement growth are set by bounds on the asymptotic entanglement capacity [10][11][12][13] and more recent small incremental entangling theorems [14][15][16][17].We show that these limits can also be used to obtain lower bounds on the execution time of Hamiltonian protocols for information transfer.This raises the question of whether a protocol can achieve optimality by saturating the bound.
Quantum state transfer studies protocols for moving qubits through a spin chain [18].Long-range interactions can be used to speed up protocols [19], but here we consider only nearest-neighbor interactions.State transfer protocols usually assume the intermediate medium to be in a known initial state [20][21][22][23] or allow it to change in an unknown or non-trivial manner [24,25].Such protocols are not directly applicable when some or all spins in the chain contain data qubits that need to be transferred or maintained.
Protocols for state reversal, also known as state mirroring [26], take steps towards addressing this issue.State reversal reverses any input state on a spin chain about the center of the chain.Specifically, with qubit labeling 1, 2, . . ., N , state reversal corresponds to the unitary up to a global phase, which is independent of the state.
State reversal is potentially a useful subroutine for the more general task of qubit routing, where we wish to apply arbitrary permutations to the qubits.Early results in this area require the state to be in the singleexcitation subspace [27] or introduce phases in the final state that depend on a non-local property such as the number of qubits in state |1 [26,28].These limitations were later removed by time-dependent protocols for state reversal [1,29,30].
In this work, we propose the first time-independent protocol for state reversal using nearest-neighbor interactions.We show that the execution time of our protocol is nearly optimal, comparable to the time-dependent protocol given in [1].However, as our protocol does not require dynamical control but only pre-engineered couplings, we expect it to be more experimentally feasible on near-term quantum systems.
Before presenting our state reversal protocol in more detail, let us elaborate on the claim that it is nearly optimal -specifically, that it has an evolution time within a factor 1.502(1 + 1/N ) of the shortest possible.For any nearest-neighbor spin Hamiltonian H, a time scale follows from a normalization that limits the strength of every two-qubit interaction but allows fast local operations.Up to local unitaries, we can write any two-qubit Hamiltonian in the canonical form [31] K := j∈{x,y,z} where µ x ≥ µ y ≥ |µ z | ≥ 0 and σ j are the Pauli matrices.We impose the normalization condition that K = j |µ j | ≤ 1 for all interactions, where • is the spectral norm.Under this normalization, a swap can be optimally implemented in time 3π/4 [32], and our protocol achieves state reversal in time where p(N ) := N (mod 2).This is equivalent in time to a swap gate circuit of depth ∼N/3.As state reversal using only swaps requires depth at least N − 1 [33], our protocol is faster than any swap-based protocol by an asymptotic factor of 3. Similarly, we can compare to other time-independent Hamiltonian protocols that use nearest-neighbor interactions: [20] implements state transfer in time N π/4 and [26] implements state reversal in time N π/2 but introduces relative phases in the state as mentioned earlier.Our time-independent protocol (and some time-dependent protocols [1,29,30]) thus improve upon these previous protocols for state transfer and state reversal except for a subleading term.We lower-bound the time for state reversal, which can generate entanglement across a bipartition, by using bounds on the asymptotic entanglement capacity in a more general model [11,13].The asymptotic entanglement capacity bounds the rate at which entanglement can be generated by any evolution of a given bipartite Hamiltonian interspersed with arbitrary local operations and classical communication (LOCC) and with arbitrary finite local ancilla spaces.We give an explicit example of entanglement generated by state reversal and lowerbound the time using the capacity of a normalized twoqubit interaction in canonical form (2), even allowing for LOCC.Nonetheless, our state reversal protocol is able to nearly saturate this bound without classical communication, without ancillas, and with only nearest-neighbor interactions throughout the chain.
We propose a state reversal protocol with Hamiltonian of the form where the coefficients J , h are engineered as follows.Letting for k ∈ N, our protocol is defined as (see also Figure 1) for all sites k, and let H := H(J , h).Apply U := e −it N H to the input state.x couplings (J k , red) and on-site σ k z fields (h k , blue) are plotted on the y-axis.Sites 0, N + 1 are ancilla qubits, which are not part of the protocol and are used purely in the analysis.
We show in the following sections that our protocol implements state reversal exactly, up to a global phase (we denote this equivalence by ∼ =).In other words, Proof and analysis of the protocol.We prove the correctness of our protocol (i.e., Theorem 2) by mapping the spin chain to a doubled chain of Majorana fermions via a Jordan-Wigner transformation, describing the action in the Majorana picture, and then mapping back to the spin picture.To help with the analysis, we extend the chain with two ancillary sites {0, N + 1} called the edge, E, and refer to the sites {1, . . ., N } as the bulk, B. We define the transverse-field Ising model (TFIM) Hamiltonian on the extended chain that reduces to H when the edge is initialized to state |++ .Similarly, we define U := e −i Ht N .Note that the operator H (and hence U ) acts trivially on |++ E , so this edge state does not change through the course of the evolution.(Our results also hold using the edge state |−− E , which is equivalent to negating the sign of the longitudinal fields in (4).)We then prove that in the Heisenberg picture, Pauli matrices on site k map to the corresponding Pauli on site N +1−k for all sites k in the chain.First, we map to the doubled chain of Majorana fermionic operators by defining at each site, where we have used the notation P We also see that (6) to take the form The Majoranas γ 0 , γ 2N +3 do not appear in the sum, since a 0 = a 2N +2 = 0.
Lemma 3. The operation U acts on the Majorana operators as Proof.For the first case, H has no overlap with operators γ 0 and γ 2N +3 , so they are stationary under evolution by H.
For the remaining cases, we make an analogy with the dynamics of the y component of the spin operator, S y , for a spin s = N + 1  2 particle (as in, e.g., [26]).The Heisenberg evolution of γ k corresponds to the rotation of the (with = 1), we can express (8) in the bilinear form H = 1 2 γ † Aγ, for the vector γ := γ 1 γ 2 . . .γ 2N +2 and the matrix A := −π/(2t N )S y expressed in the S z basis.Using the Majorana commutation relations, we have γ = i[ H, γ] = 2iAγ, so γ(t) = e 2iAt γ(0).The Heisenberg evolution of γ k under H for time t N is exactly analogous to the (Schrödinger) time evolution of the state |s, k − s − 1 under S y for time π.A π-rotation under S y maps and correspondingly, Note that (11) can easily be verified for a spin-1/2 particle.Similarly, a spin-s particle may be viewed as a system of 2s spin- 1  2 particles with maximal total spin.In this picture, a π-rotation under S y corresponds to independent π-rotations of each small spin.Since the state |s, k − s − 1 is represented by a permutation-symmetric state with k − 1 up spins, the π-rotation maps it to a state with 2s − (k − 1) up spins and introduces a phase (−1) for each up spin, which is precisely (11).
Due to the signed reversal of the Majoranas in Lemma 3, the parity string P [0,k] = i b+1−a 2b+1 j=2a γ j is (with the exception of γ 0 ) reflected about the center of the chain with an overall phase that exactly cancels when the product is reordered by increasing site index.The invariance of the edge Majoranas is crucial, as it provides a phase factor that cancels the state-dependent phases when we revert to the spin picture.In particular, we have the following lemma.

Lemma 4. The operation U acts on the parity strings as
Proof.Applying Lemma 3, we have where we reordered the product and used From the Majorana anti-commutation relations and ( 7), the result follows.
Now we prove the main theorem.
Proof of Theorem 2. U ∼ = R holds iff all bulk observables on the chain transform identically under U, R. For any operator O k supported on bulk site k ∈ {1, . . ., N }, we show that (Henceforth we drop the edge subscript E.) By (7) and Lemmas 3 and 4, σ k x is mapped to Next, we use Lemma 4 to show that σ k z is mapped to All other observables can be written in terms of the onsite Pauli operators σ k x , σ k z , so U is identical to R, up to global phase.
where H AA BB is the Hilbert space of the bipartite systems A and B with arbitrarily large ancilla spaces A and B , respectively.In particular, for a Hamiltonian of the form σ x ⊗ σ x , [10,11] showed that (21) This is tighter than the more general small incremental entangling bound E H ≤ c H log 2 d = 2 for the conjectured c = 2 [14] (best known c = 4 [16]) and where the smallest dimension of A or B gives d = 2. Since E is invariant under local unitaries, a direct corollary is that We now show that Protocol 1 is close to the shortest time possible.
Theorem 5.It holds that Proof.We prove the time lower bound via an upper bound on the rate of increase of entanglement across a cut in the center of the chain (allowing differences of one qubit for odd N ).Designate the left half of the cut as subsystem A and the right half as subsystem B. A consists of subsystem A given by the qubit at site N/2 adjacent to the cut, and subsystem A consisting of the remaining qubits to the left of the cut as well as a finite but arbitrary number of ancilla systems that are not part of the chain.Similarly, B consists of subsystem B, the qubit at site N/2 + 1, and B , the remaining qubits in the right half with an arbitrary finite number of ancilla.
Consider Hamiltonians of the form H(t) = K(t)+ K(t) specifying the evolution of the AB system, where K(t) is a two-qubit Hamiltonian supported on systems AB (i.e., the cut edge), while K contains terms supported on AA or BB but not the cut edge AB.For brevity, we drop the time parameter t even though we allow the Hamiltonian to be time-dependent.We assume that K is expressed in canonical form (2) due to equivalence under local unitaries.Aside from its support, we make no assumptions about the form of K (so the resulting bound is more general than nearest-neighbor interactions).We call H satisfying these conditions divisible and also call protocols using divisible Hamiltonians divisible.
Observing that E H is the supremum over a time derivative of the von Neumann entropy of ρ = Tr B |ψ ψ|, we have The reduced density matrix ρ has time evolution We substitute H = K + j∈{x,y,z} µ j σ j ⊗ σ j in the commutator and substitute the time-dependence of ρ into (23).By linearity of the trace and sublinearity of the supremum, we get where we observe that E K = 0 since K does not have support across the cut, and use the normalization condition j |µ j | ≤ 1.This bound holds for all divisible Hamiltonians H, with nearest-neighbor Hamiltonians as a special case.
The entanglement generated by any divisble protocol can now be bounded in time.We observe that if the protocol contains local measurements then these cannot increase entanglement E(|ψ ) and that feedback may be viewed as a particular time-dependence of H conditioned on measurement outcomes.Therefore, (25) bounds the total increase in entanglement across bipartition AB over a time t * by for any initial state |ψ(0) acted on by a divisible protocol and LOCC.Finally, we give an explicit bound on the worst-case time of divisible state reversal protocols by specifying an initial state.Let the system start in the product state |φ A ⊗ |φ B where each qubit forms a Bell state with a local ancilla not part of the chain.Clearly, E(|φ A ⊗ |φ B ) = 0. We perform a reversal R on the chain and get the state |ψ AB := R(|φ A ⊗ |φ B ), which is maximally entangled, i.e., E(|ψ AB ) = N .Then, (26) gives the bound on any divisible state reversal protocol.Comparing this to our protocol time (3), we have Discussion.The time-dependent protocol in [1] is closely related to our time-independent protocol, and both can be described within the same framework (see appendix).In the time-dependent case, the state is evolved alternately under two restrictions of the Hamiltonian (4): H(1, 0) (uniform Ising) and H(0, 1) (uniform transverse field), each for time π/4, for a total of N + 1 rounds.In the Majorana picture, these Hamiltonians carry out a simultaneous braiding of neighboring Majoranas along even (resp.odd) edges of the doubled Majorana chain.The resulting map matches Lemma 3 exactly, implying that the two protocols are identical at the level of Majorana operators.Indeed, any protocol achieving the map in Lemma 3 is guaranteed to implement state reversal.
In fact, as shown in the appendix, there is an infinite family of nearest-neighbor, time-independent Hamiltonian protocols for state reversal that generalizes Protocol 1.The family is parameterized by a non-negative integer m, with modified σ k x σ k+1 x coupling J (m) k ∝ (2N + 1 − 2k + 4m) (2k + 1 + 4m) and unmodified σ k z field strength.Protocol 1 corresponds to the special case of m = 0.By choosing large m, the coupling strength can be engineered to be nearly uniform throughout the chain, which may be a desirable feature in experimental implementations of the protocol [28].
State reversal implements a specific permutation of qubits in a spin chain faster than naively possible using swaps.More generally, we would like to know how we can perform qubit routing on a spin chain faster than possible naively (for example, by using fast state reversal as a subroutine).Moreover, while our lower bound shows that sublinear scaling is not possible for qubit routing on a spin chain, it is still an open question whether a superconstant advantage over routing using swaps is possible with other interaction structures.Answers to these questions have applications in circuit transformations for quantum architectures [5], where qubit routing is a key subroutine. A

APPENDIX
Time-dependent protocol for reversal.In this section, we give a simple analysis of the timedependent protocol given in [1,29] using our methods.The strategy is to prove that this protocol satisfies Lemma 3 from the main text.Lemma 4 and Theorem 2 are then automatically satisfied.First, we reintroduce the protocol using our notation.Protocol 6.Let H h := H(0, 1) and H J := H(1, 0), where 1 = (1, 1, . . ., 1) and 0 = (0, 0, . . ., 0).Explicitly, FIG. Apply V := e i π 4 H h e i π 4 H J N +1 to the input state.
As in the main text, we extend the chain with two ancillary sites {0, N + 1} that constitute the edge E. The unitary V extends to an operator V := 1 E ⊗ V on the extended chain.Then the following lemma holds.Lemma 7. The operation V acts on the Majorana operators as Proof.We use (7) to write V as a product of alternating π/4-rotations under two Hamiltonians H J = i N k=0 γ 2k+1 γ 2k+2 and H h = i N k=1 γ 2k γ 2k+1 .Since e −π/4γiγj is a braiding unitary that maps γ i → γ j , γ j → −γ i , γ k =i,j → γ k , it follows that the operator e i π 4 H h braids nearest-neighbor Majoranas along all odd edges of the chain (except the first and last edge), while e i π 4 H J braids along the even edges.Therefore, alternating π/4 rotations under H J and H h implement an even-odd sort [34] on the chain, as shown in Figure 2. Accounting for sign changes, the Majoranas map as follows: γ k → (−1) k+1 γ 2N +3−k , while γ 0 , γ 2N +3 remain unchanged.
Infinite family of Hamiltonians for state reversal.Reference [28] shows that there is an infinite family of XY Hamiltonians that generalize the protocol introduced in [26].In fact, Protocol 1 is also a special case of an infinite family of protocols parameterized by a single non-negative integer m, as given below.
Apply U (m) := e −iH (m) to the input state.
The protocol modifies only the couplings J (m) k as a function of m, while the field terms h (m) k = h k are invariant with m.Note that U (0) = U , so Protocol 1 is indeed a special case of Protocol 8.For convenience, we have rescaled the coefficients so that the evolution time is 1.To prove the correctness of this family of protocols, write the Hamiltonian H (m) in terms of Majorana fermions obtained by Jordan-Wigner transformation on the spin chain (extended to edge sites {0, N + 1}).We have where γ = γ 1 γ 2 • • • γ 2N +2 and A (m) is a (2N + 2) × (2N + 2) tridiagonal matrix with entries As before, the Heisenberg evolution of the Majoranas under H (m) is given by γ(t) = e 2iA (m) t γ(0).Lemma 3 shows that the operator e 2iA (0) implements reversal.Here we show that e 2iA (m) = e 2iA (0) for all m, which implies that U (m) implements state reversal for all m.We state the following lemma (due to [35,36]) on the spectrum of A (m) .
Proof.The first claim follows from [35].Via a transformation of the off-diagonals that preserves the spectrum, A (m) can be converted to a matrix B(n, a) of Sylvester-Kac type for n = 2N +1, a = 4m.As shown in [35], the eigenvalues of B(n, a) are given by the formula λ ±,j = ± π 4 |2j + 1 + a| for j ∈ {0, . . ., n}, and the first claim follows.
When normalized so that all two-qubit terms are bounded by unity in spectral norm, H (m) implements state reversal in time t .Setting m = N + 1, for example, yields a variation in the couplings on the order of 3% for any N , and gives reversal in time 5N π/4.

FIG. 1 .
FIG.1.The state reversal operation R (depicted by arrows) and an illustration of our time-independent protocol to implement it.The nearest-neighbor σ kx σ k+1 [a,b] := b j=a (−σ j z ) for the Jordan-Wigner parity string between sites a and b.The γ k are Hermitian and satisfy the Majorana anti-commutation relations {γ j , γ k } = 2δ jk .
Time lower bound.We now prove a lower bound on the optimal time, t * , to implement state reversal using normalized local interactions.Let the entanglement entropy between systems A and B of a bipartite state |ψ AB be E(|ψ ), defined as the local von Neumann entropy S(ρ) := − Tr[ρ log 2 ρ], for ρ = Tr B [|ψ ψ|].Then, the asymptotic entanglement capacity of a Hamiltonian H that couples systems A and B was shown to equal[13] .B. and A.G. acknowledge support by DoE ASCR Quantum Testbed Pathfinder program (award number de-sc0019040), DoE ASCR FAR-QC (award number de-sc0020312), DoE BES Materials and Chemical Sciences Research for Quantum Information Science program (award number de-sc0019449), NSF PFCQC program, AFOSR, ARO MURI, ARL CDQI, and NSF PFC at JQI. E.S. and A.C. acknowledge support by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Quantum Testbed Pathfinder program (award number de-sc0019040) and the U.S. Army Research Office (MURI award number W911NF-16-1-0349).

4 . 2 .
Therefore, the time cost increases linearly in m and is minimal for Protocol 1 where m = 0. Next, observe that if we choose 4m N , the variation in coupling coefficients J Therefore, the parameter m quantifies a trade-off between reversal time and the non-uniformity of J (m) k 2. Time-dependent reversal protocol for N = 2 (with two edge ancillas).For any bulk state |ab 12 (with edge state |++ E ), alternating π/4 evolutions under H2, H1 are applied a total of 2N + 2 times.Each step braids neighboring Jordan-Wigner Majoranas; the right-movers (red) keep the same sign while the left-movers (blue) gain a minus sign.The edge Majoranas γ0, γ7 are unchanged (a crucial feature that ensures the correct parity phases), while the intermediate Majoranas undergo reversal of position with alternating sign.The final state in the bulk of the chain is |ba 12 .