Classical Variational Optimization of Gate Sequences for Time Evolution of Large Quantum Systems

Classical Variational Optimization of Gate Sequences for

Time Evolution of Large Quantum Systems

Refik Mansuroglu

, Samuel Wilkinson

, Ludwig N¨

utzel

, and Michael J. Hartmann

The simulation of time evolution of large quan-tum systems is a classically challenging and often intractable task, making it a promis-ing application for quantum computation. A Trotter-Suzuki approximation yields an imple-mentation thereof, where a certain desired ac-curacy can be achieved by raising the gate count adequately. In this work, we introduce a variational algorithm which uses solutions of classical optimizations to predict efficient quantum circuits for time evolution of large quantum systems. Our strategy improves on the Trotter-Suzuki ansatz in accuracy and gate count by several orders of magnitude. In a NISQ-friendly setting, we can either signifi-cantly reduce the approximation error while using the same amount of gates as the Trotter-Suzuki ansatz or achieve a comparable accu-racy with significantly fewer gates. We find, that our strategy outperforms the Trotter-Suzuki benchmark in translation invariant sys-tems as well as for open boundaries.

1

Introduction

Quantum simulation of real time evolution represents a very natural application of quantum computation. In particular non-equilibrium dynamics is known to lead to highly entangled states [18,5] and is therefore a very strong candidate for a quantum advantage, i.e. an application where quantum computers can com-pute quantities of interest that are not accessible to classical architectures [1]. Given a time-independent Hamiltonian H for a closed, physical system of N qubits, one can define the time-evolution operator as the unitary

Uex= exp (−iHt). (1)

This operator propagates a quantum state |ψ(t0)i

along a time interval t to |ψ(t0+ t)i. The

dimen-sion of the underlying Hilbert space grows exponen-tially in N , which makes simulation of large quan-tum systems intractable for classical computation. A

Refik Mansuroglu: [email protected]

Samuel Wilkinson: [email protected]

quantum computer, however, can directly operate on the N qubits with quantum gates. As we are un-likely to have a quantum gate of the form of Eq. (1) at hand, when H contains several non-commuting terms, a decomposition into experimentally accessible quantum gates is needed. One such decomposition can be achieved by a Trotter-Suzuki approximation [16, 15,9].

Since quantum simulation on modern noisy inter-mediate scale quantum (NISQ) [13] devices suffers from gate errors, variational methods have been put forward to reduce the total gate count and circuit depth [14, 11, 4, 12]. While a Trotter-Suzuki de-composition already yields a good approximation to a Hamiltonian-generated unitary time evolution, it has been shown that variational methods can improve on this approach [7, 3, 2] as one can trade off the im-provement of approximation accuracy against the gate count.

Here we introduce an optimization method to find initial-state-agnostic variational quantum circuits for quantum simulation. Given a translationally invari-ant Hamiltonian, we find a generalization scheme to obtain the quantum circuit of a large system from the optimal solution of a small, classically tractable system. One can think about our approach as gluing together several copies of the time evolution of a small system, which have been optimized.

When translational invariance is broken by open boundary conditions, we propose a variant of the glu-ing method which yields the time evolution of the cor-responding open boundary system. This scenario is favored by modern NISQ devices which would require a number of SWAP operations to implement periodic boundaries.

We illustrate the method by performing a quantum simulation of a one-dimensional Transverse Field Ising model. Generalizations to two-dimensional qubit grids are straight forward. For small systems for which we can compare the quantum circuit with ex-act time evolution, our variational approach improves upon the Trotter-Suzuki decomposition in terms of both accuracy and gate count. More interestingly, we are able to suggest quantum simulation circuits for larger systems, which outperform the Trotter-Suzuki ansatz in a similar way.

The improvement we find is significant in

friendly settings as well. Here the simulated time at fixed gate count can be significantly larger or the re-quired gates for a fixed time span can be significantly lower than for Trotter-Suzuki sequences. For realis-tic gate fidelities, this in parrealis-ticular allows for time simulations that cannot be performed with suitable accuracy with a Trotter-Suzuki ansatz.

2

Variational Optimization

We consider a p-local quantum many-body Hamilto-nian H which consists of ` terms H1, ..., H` and acts

on N qubits. The parts {Hi}i=1,...,l of the

Hamilto-nian are chosen such that each Hi is the generator

of a quantum gate that can straight-forwardly be im-plemented on the considered hardware1_{. The unitary}

time evolution generated by H can be approximated by a Trotter decomposition [16], Uex= exp −it l X i=1 Hi ! = " ` Y i=1 exp −it m Hi #m + O t 2 m2  , (2)

where m is often referred to as the order of Trotter ap-proximation. The error of this approximation is given by the exponential of higher order nested commuta-tors between H1, ..., H`, which can be seen through

the Baker-Campbell-Hausdorff Lemma. These can be compensated by a Suzuki approximation [17]. In some cases, a second order Suzuki approximation,

Utrot= " ` Y i=1 exp −it 2mHi  1 Y i=` exp −it 2mHi #m , (3) can be considered without significant increase in gate count and circuit depth. This way, the error of Trotter-Suzuki approximation is of order O_mt33

 . We will stick to first order Suzuki in the main text of this paper and discuss a second order approach in Ap-pendixA.

It has been shown [7] that variational methods can improve on Trotter-Suzuki ans¨atze for quantum simulation in general. For this, a gate sequence of parametrized unitaries is considered:

Uvar(~θ) = U1(θ1) · U2(θ2) · · · (4)

Our method will be independent of the particular ansatz Uvar. Here, we use the Trotter-Suzuki

approx-imation from Eq. (2) as a starting point, as it en-sures an accuracy of at least as good as Trotter-Suzuki approximation. In order to exhaust the potential of the ansatz from Eq. (2), we will substitute the co-efficients for each of the Hamiltonians Hi at each of

the m sequences by a variational parameter θi,j with

i ∈ {1, ..., `} and j ∈ {1, ..., m}. The resulting ansatz

reads Uvar(~θ) = m Y j=1 " ` Y i=1 exp  −i 2θi,jHi # . (5)

We suggest to find a set of optimal parameters {θi,j}

for which Uvar approximates the exact time evolution

Uexbest. Since the Trotter approximation of Eq. (2)

is included in the possible choices for the parameters

θi,j, the optimal choice for the θi,j typically leads to

a better approximation and can only improve on the Trotter approximation. Note that also for the vari-ational ansatz, there is an approximation order m which controls the number of parameters. There are thus ` · m different parameters to tune.

The optimal parameters are then found by the min-imization of a cost function measuring the distance of unitaries. We suggest a measure of distance induced by the Frobenius norm

C(~θ) = ||Uvar(~θ) − Uex||2F = X a,b |Uvarab− Uexab| 2_, (6) where Ua_b denotes the elements of a matrix represen-tation of the operator U . Here, the exact operator of time evolution Uexserves as a benchmark which is

ag-nostic of an initial state. In a similar fashion, we can also define a measure of distance between the Trotter-Suzuki ansatz and exact time evolution. We will later discuss the optimization of a one-dimensional Trans-verse Field Ising model, but for now let us assume, we have found a set of parameters ~θoptsuch that C(~θ) as

defined in Eq. (6) is minimal.

Note that Eq. (6) involves the full time evolution operator which is not accessible on a quantum puter, but has to be calculated on a classical com-puter. This means that we do not have any quantum advantage yet, if we employ Uvar on a quantum

com-puter. In order to arrive at a meaningful application for quantum computation, we can apply the optimal parameters to a larger, yet unseen system. We ad-dress how to do this in the following.

1_{Depending on the p-locality of H and on the quantum}

a)

c)

b) d)

... ...

Figure 1: a) One dimensional spin chain of 4 and 8 qubits with periodic boundary conditions. The 8 qubit chain can be looked at as two 4 qubit chains glued together. When constructing the Hamiltonian of the larger system, one recovers two-qubit interactions of the small building block several times. Two such interactions are indicated by green and red lines. b) Sketch of the corresponding quantum circuit. Also here, two copies of a circuit on 4 qubits each are glued together to an 8-qubit circuit. The rectangles depict 2-qubit gates corresponding to the 2-qubit interactions before. The gates colored in red and green correspond to the interaction scheme of a). The fading red color at the top and bottom indicates that both qubits are subjected to the same gate closing the boundary to a closed loop. c) One dimensional spin chain of 8 qubits with open boundary conditions. By cutting the open boundary system in half and filling an arbitrary number of blocks with periodic boundary conditions (purple block, cf. a)), it is possible to construct the Hamiltonian of a large system with open boundary conditions. The interaction term at the cutting point (red) is replaced by the interaction term at the periodic boundary of the first and last purple block, respectively. The other interaction terms are just being copied as before (cf. green line for instance). d) Sketch of the corresponding quantum circuit. The left and right boundary is being glued together with many copies of a circuit for periodic systems which are represented by the purple rectangular placeholder. The gates colored in red and green correspond to the interaction scheme of c).

3

Gluing Operators Together

Hamiltonians of many body quantum systems often enjoy translational symmetries, such that larger sys-tems look similar to smaller syssys-tems, locally. We in-troduce a method which exploits this fact and gen-eralizes solutions of variational optimizations to large quantum systems.

Since modern quantum hardware, however, natu-rally comes with open boundary conditions for the implementable gates which disrupt translational in-variance, we also extend the method by open bound-ary conditions.

3.1

Translationally Invariant Systems

Translationally symmetric systems can be scaled up by successive repetition of many unit cells. Fig. 1 a), for instance, shows a one dimensional spin chain with periodic boundary conditions. If we consider nearest

neighbor interactions, then the interaction between the first two qubits is the same as the interaction be-tween any other neighboring pair of qubits. From this manifest translational invariance, we are urged to gen-erate ans¨atze for larger quantum systems by copying the quantum gates of the small circuit. To this end, we apply the quantum gates, which before acted on qubits N and 1, now to qubits N and N + 1 and sim-ilarly to 2N and 2N + 1 and so on. The final glued variational circuit then reads

U_var(1,k·N )(~θ) =

k−1

Y

p=0

U_var(pN +1,pN +N +1)(~θ), (7)

where Uvar(a,b)denotes the variational ansatz which acts

on the qubits a, a + 1, ..., b and qubit kN + 1 ˆ= qubit 1.

varia-tional ansatz and illustrate the gluing method with the Transverse Field Ising model. For N qubits, this model is given by the Transverse Field Ising Hamilto-nian HS= N X i=1 (JzZi⊗ Zi+1+ hxXi) , (8)

with the Pauli matrices X, Z and the coupling con-stants Jz and hx. We denoted a Pauli Z operator

acting on the ith qubit by Zi. Also ZN +1= Z1 (and

analogously for X). In the gluing procedure, we copy all the terms k times and map the boundary terms to bulk terms

ZN ⊗ Z17→ ZN ⊗ ZN +1

Z2N ⊗ ZN +17→ Z2N ⊗ Z2N +1

...

This procedure is also sketched in Fig. 1 b). Ulti-mately, we end up having the Transverse Field Ising Hamiltonian for k · N qubits.

This process might seem redundant at first. How-ever, if we recall the variational method to improve on trotterized time evolution, we have just derived a generalization scheme which allows us to use the pa-rameters ~θoptfound for the N qubit system to suggest

a decomposition of quantum gates for the simulation of the k · N qubit system. This is done by, again, breaking the periodic boundary conditions and gluing

k copies of the Hamiltonian but now with the

corre-sponding variational parameters instead of interaction constants. The resulting gate sequence then reads

Uvar(~θ) = m Y j=1 "k·` Y i=1 exp  −i 2θi mod N,jHi # . (9)

This sequence can now be implemented on a quantum computer. From the translational invariance of the system, we expect the advantage in accuracy which is gained within classical optimization of the N qubit system to withstand the gluing process.

It becomes apparent that the gluing method is iden-tical to Trotterization on a coarser level. Given the Hamiltonian HL of the large system, exp(−itHL) is

split into the k blocks of the small systems. In a standard Trotter-Suzuki approach, these can then be split successively into their Trotter-Suzuki decompo-sition as described in Eq. (3). As a result, there is no difference in a Trotter-Suzuki decomposition of exp(−itHL) and gluing k copies of Trotter-Suzuki

decomposition of exp(−itHS) with HS denoting the

Hamiltonian of the small system.

3.2

Open Boundary Conditions

On some quantum hardware, such as superconduct-ing circuit devices, the implementation of periodic boundary conditions involves additional use of SWAP

gates. Thus, the simulation of physical systems with open boundary conditions is better suited for such de-vices, particularly under NISQ conditions. With an eye to an implementation of the gluing method, we thus propose an adaption scheme to physical systems with open boundary conditions.

Instead of only gluing together identical blocks of

N qubits, as discussed above, we introduce a second

block of N qubits described by a Hamiltonian which breaks the periodic boundary conditions

HS,o= N −1

X

i=1

(JzZi⊗ Zi+1+ hxXi) . (10)

The additional subscript o denotes that we are consid-ering a system with open boundary conditions. The Hamiltonian in Eq. (10) generates time evolution which can be approximated by the operators Utrot,o

and Uvar,o(~θ) in an analogue way as before. When the

optimal parameters ~θoand ~θ for both, open and

peri-odic boundaries, have been found, the final circuit for time evolution of a system of (k + 1) · N qubits with open boundaries can be constructed the following way

U_var,o(1,(k+1)·N )=U(1,

N 2)

var,o (~θ1,o)Uvar(~θ)

× U(kN +N2+1,(k+1)N )

var,o (~θ2,o) (11)

where we used Uvar(~θ) from Eq. (7) for the time

evolu-tion of the bulk. Furthermore, we split the optimized parameter set of the open boundary N qubit simula-tion in half, i.e.

θk_1,o= θok, ∀k ∈ {1, ...,

N

2} (12)

and θk_2,o= θ_ok, ∀k ∈ {N

2 + 1, ..., N } (13) in order to use them for the time evolution of the left (or respectively right) boundary. For the sake of simplicity, we assume N to be even. The construction in Eq. (11) is illustrated in Fig. 1 c-d).

We will now show the effectiveness of our method in numerical experiments.

4

Transverse Field Ising Model

We studied the Transverse Field Ising model on a one dimensional, closed spin chain to test the performance of the variational gluing method in comparison to the Trotter-Suzuki ansatz. For this, we simulate the exact time evolution Uex generated by the Hamiltonian in

Eq. (8), as well as the Trotter-Suzuki Utrotand

varia-tional sequence Uvar and compare the cost values. To

Figure 2: (a) Cost values C(~θopt) of the time evolution of a 6-qubit Transverse Field Ising model with coupling constants

Jz ∈ [0.1, 1] and hx ∈ [−1, 1]. The variational ansatz after

optimization (left) improves on the Trotter-Suzuki approxi-mations (right). (b) Cost values of the time evolution under a 12-qubit Transverse Field Ising model. Using two copies of the parameters from optimization of the 6-qubit system for the variational ansatz (left) still improves Trotter-Suzuki approximation (right). In all cases, we used an aproximation order of m = 3 and a simulation time t = 0.3.

We also challenge the simulation of larger simu-lation times by repeated application of the gate se-quences. For optimization we have used the Adam gradient descent method [8] 2_.

2_{In all the optimizations, we have used a maximal number}

of traning steps of 104_{, a learning rate of 10}−4_{and the}

hyper-parameters β1 = 0.9, β2 = 0.99 to control the decay rates of

the gradient and squared gradient average.

Figure 3: Normalized cost values for the time evolution under

a Transverse Field Ising Hamiltonian with Jz = 0.25, hx =

1, t = 0.3 for different system sizes and gluings. In addition to the improvement on equal sized circuits, the variational method also beats Trotter-Suzuki approximations of double the gate count in terms of cost value. The circles depict single blocks (k = 1), squares and triangles indicate the cost values of two (k = 2) and three (k = 3) blocks which were glued together. Optimization has always been performed on a single block of N qubits.

4.1

Performance

There is an ambiguity in the choice of the ordering of {Hi}i=1,...,k·`, which leads to different operators when

doing Trotterization. We fixed the ordering for the sake of an efficient implementation to

Uvar(~λ, ~ω) = m Y j=1 " ` Y i=1 i odd

e−iλi,jZiZi+1

×

`

Y

i=1 i even

e−iλi,jZiZi+1 ` Y i=1 e−iωi,jXi # . (14)

The gates we use can be implemented by powers of

ZZ and X. In Fig. 2, we show the resulting cost values C(~θopt) from the optimization of Uvar on a

6-qubit spin chain and compare these to the Trotter-Suzuki ansatz. With the variational method, we can improve on Trotter-Suzuki by three orders of magni-tude in terms of the cost value. Interestingly, also if we consider two copies of the 6-qubit chain, as de-scribed above, we stay with a similar improvement.

The total number of gates G and the circuit depth

D, in general, is given by the approximation order m.

In particular, for the Trotter-Suzuki decomposition of the Transverse Field Ising model, G and D read

Figure 4: Normalized cost values for up to 100 repetitions of the time evolution under a Transverse Field Ising Hamiltonian

with Jz = 0.25, hx = 1, t = 0.3 of two copies of a 6-qubit

system. Although the variational ansatz beats Trotterization to an order of m = 6 for a single time step, it becomes less performant for many time steps.

against Trotter-Suzuki approximation of larger m for the price of higher gate count and circuit depth in Fig. 3. As our cost function (6) is sensitive to the dimension of the Hilbert space, we normalize it by the norm of the identity ||1||2F = 2

kN_{. For various number}

of qubits, our variational ansatz is able to compete against higher order Trotter-Suzuki approximations. This implies that we can reach both, a better accuracy and a reduction of gate count and circuit depth. Fig.

3 also shows that the cost function values for glued systems (k > 1) hardly differ from the single block systems.

Since we determine Uvar via a state agnostic

opti-mization, we can apply it over and over to the time evolved state. In this way we can cover larger sim-ulation times t by feeding the state through the cir-cuit several times. In Fig. 4, we did this for the previously discussed 12-qubit system. While the cost values stays approximately constant for the Trotter-Suzuki approximation, the variational ansatz gets less performant over time.

The cost function in Eq. (6) is a measure of how well time evolution is approximated in the full Hilbert space and is therefore only an upper bound to the error of a given observable

| hψ|Uex†nOU

n

ex|ψi − hψ|Uvar†nOU

n

var|ψi |

≤ κ||O||∞||Uvar− Uex||F, (16)

with the operator norm ||.||∞ and some constant κ.

This means that the cost values, although yielding an upper bound for the performance, do not strictly determine the hierarchy of the deviations in an

ex-Figure 5: Error of the dynamics of the two-point spin correla-tion of a random input state for the system considered in Fig. 4. The variational algorithm competes very well with m = 6 Trotter-Suzuki decomposition also for large time steps.

periment. The error is then dependent on the precise observable and on the chosen initial state. In Fig. 5, we illustrate this phenomenon. There, we simulate the two-point correlation of two neighboring spins on a 12-qubit spin chain, e.g. hσz

6σz7i. As we are

simulat-ing the full unitary time evolution operator, we can freely choose the initial state. We find that for a ran-domly chosen input state, the two-point correlation hσz

6σz7i predicted by Uvar lies closer to the exact

dy-namics than any of the benchmarks, although m = 6 Trotter-Suzuki approximation has a lower cost value than our variational circuit for longer times.

Note that the gluing procedure is not sensitive to entanglement as the time evolution operator is agnos-tic of the initial state. States of high entanglement are particularly interesting, since the Trotter error of localized physics has been found to be independent of simulation time [6].

4.2

NISQ devices

Current and near term quantum hardware in the so-called NISQ era will not be able to run quantum er-ror correction, implying that all gates will have non-vanishing errors or infidelities. When quantum sim-ulation is performed on real devices, there are two sources of error: algorithmic error stemming from the approximation of the time evolution operator (i.e. the error that would persist even on a fully error-corrected quantum computer), and circuit error stemming from the hardware itself, including gate errors, dephasing and decoherence. To account for both sources of er-ror, the performance of algorithms on NISQ devices can be evaluated and compared using the infidelity be-tween the result of the quantum simulation and the exact result for a given random input vector ψ drawn uniformly from C2(kN )

.

For the variational ansatz this reads,

Figure 6: Algorithmic fidelity for a random input vector for up to 100 repetitions of the time evolution under a Transverse

Field Ising Hamiltonian with Jz = 0.25, hx = 1, t = 1.0 of

two copies of a 6-qubit system. Also here, the variational ap-proach gets overtaken for longer times, but still beats Trotter order m = 4.

with an analogous expression for the Trotter-Suzuki approximation. In the absence of noise, we refer to this quantity as the algorithmic infidelity, as it quan-tifies algorithmic error. The algorithmic infidelity is upper bounded by the cost value in the following way

C(~θ) = ||Uex− Uvar||2F ≥ ||(Uex− Uvar(~θ))ψ||22

≥ I(Uexψ, Uvar(~θ)ψ). (18)

This indicates that the cost function is a faithful mea-sure of the performance of the algorithm on quantum hardware.

In Fig. 6, we plot the algorithmic infidelity for 100 repetitions of the optimized gate sequence. At the same time, we increased the simulation time of a sin-gle repetition to t = 1.0 trading larger total simula-tion time against a larger absolute algorithmic error of the quantum circuit. Again, we recover an improve-ment on m = 6 Trotter-Suzuki with the variational circuit for a single time step and the improvement gets smaller when repeating the circuit several times. It falls behind m = 5 Trotter-Suzuki decomposition after 100 repetitions, but stays better than m = 4 for the entire covered time range. Hence, even after 100 repetitions, the variational algorithm yields an improvement on both precision and gate count.

On a NISQ device, there will also be infidelities from gate imperfections which we want to compare to the algorithmic infidelity. Consider a simple error model which estimates the circuit fidelity as the prod-uct of the individual gate fidelities for every gate in the circuit. The overall circuit infidelity then reads

I(Uexψ, E(ψ)) = 1 − pG, (19)

with E being a quantum channel which implements a perfect time evolution Uex, but suffers from a gate

Figure 7: Infidelity of the output state due to algorithmic er-ror compared to infidelity due to repeated gate erer-rors from a gate fidelity of 99.9%. The algorithmic error corresponds to the time evolution of the Transverse Field Ising model with

Jz= 0.25, hx= 1.0 and t = 2.5. The large value of t raises

the overall achievable simulation time when keeping the total gate count fixed, but also raises the algorithmic error signif-icantly. A NISQ device with average gate fidelity of 99.9% can resolve the improvement of the variational approach over Trotter-Suzuki.

infidelity of p for each gate. Here we use p = 0.1% (which is in the range of modern gate infidelities [10]) as an estimate for the infidelity of a single two-qubit gate.

In Fig. 7, an m = 3 Trotter-Suzuki decomposi-tion for the time evoludecomposi-tion of a Transverse Field Ising model with Jz = 0.25, hx = 1.0 has been optimized

for a single time step of t = 2.5. The large simulation time raises the circuit infidelity into a regime which is comparable to gate infidelities on NISQ devices. In our example, the algorithmic infidelity induced by a Trotter-Suzuki decomposition of m = 3 will yield a random output, whereas the quality of the varia-tional time evolution is only limited by the gate infi-delities. As a consequence, Trotter-Suzuki decompo-sition is not suited for simulating this time evolution, whereas our method is capable of doing this.

4.3

Open Boundary Conditions

Figure 8: Normalized cost values for up to 100 repeti-tions of the time evolution under a 12 qubit Transverse Field Ising Hamiltonian with open boundary conditions and

Jz = 0.25, hx = 1, t = 0.3. The 12 qubit system has been

constructed as described in Fig. 1 d)from two 6 qubit blocks,

one with open, one with periodic boundaries. The striking

similarity of the plot with Fig. 4shows that the additional

error introduced through the combination of the two different optimized clocks is small.

up to 100 time steps.

From the fact that the behavior of these cost values for an open boundary system is strikingly similar to the periodic boundary analogue in Fig. 4, we deduce that the additional error introduced by the gluing of the two optimal solutions is negligibly small. Yet, a slight increase in cost is observable.

5

Summary

We presented a variational method for finding accu-rate decompositions of the time evolution operator on a quantum circuit. We then introduced a method for using the optimal parameters found on a classically-tractable system to predict optimal parameters for larger systems by gluing together gate sequences. We further showed that this gluing method can be applied

to periodic as well as to open boundary conditions. The latter form an interesting class of problems for the implementation on NISQ hardware which often comes with open boundary conditions.

We find that our method outperforms the Trotter-Suzuki approximation, either reducing the necessary circuit depth and number of gates while keeping the same approximation error or achieving much higher accuracy at the same gate count.

Our results open up a highly versatile way of vari-ational quantum simulation. Since we have shown that it is sometimes sufficient to maximize the simu-lation quality locally, possible limitations of this glu-ing scheme are worth investigatglu-ing further. We also expect to gain further insights into the scope and per-formance of our method from investigations of the relations between the range of correlations in simu-lated quantum states and the size of the blocks con-sidered in the classical optimization of the variational paramters. Since the optimization is however state-agnostic, depending only on the Hamiltonian of the system, it is expected to perform well at simulat-ing the time evolution of highly-entangled states with long-ranged correlations.

Another way of optimizing the Frobenius distance between Uex and Uvar can be achieved by

random-ized benchmarking. This way, the cost value is only approximated, but opens up a genuine quantum op-timization.

Acknowledgments

This work received funding from the European Union’s Horizon 2020 research and innovation pro-gram under Grant Agreement No. 828826 ”Quromor-phic.”. We thank Petr Zapletal and Timo Eckstein for valuable feedback and discussions and Naeimeh Mohseni for feedback on the manuscript.

Code Availability

The code used to generate the numerical results pre-sented in this paper can be made available upon rea-sonable request.

References

[1] F. Arute, K. Arya, R. Babbush, D. Bacon, J. Bardin, R. Barends, R. Biswas, S. Boixo, F. Brandao, D. Buell, B. Burkett, Y. Chen, J. Chen, B. Chiaro, R. Collins, W. Courtney, A. Dunsworth, E. Farhi, B. Foxen, A. Fowler,

McClean, M. McEwen, A. Megrant, X. Mi, K. Michielsen, M. Mohseni, J. Mutus, O. Naa-man, M. Neeley, C. Neill, M. Y. Niu, E. Os-tby, A. Petukhov, J. Platt, C. Quintana, E. G. Rieffel, P. Roushan, N. Rubin, D. Sank, K. J. Satzinger, V. Smelyanskiy, K. J. Sung, M. Tre-vithick, A. Vainsencher, B. Villalonga, T. White, Z. J. Yao, P. Yeh, A. Zalcman, H. Neven, and J. Martinis. Quantum supremacy using a pro-grammable superconducting processor. Nature, 574:505–510, 2019.

[2] S. Barison, F. Vicentini, and G. Carleo. An ef-ficient quantum algorithm for the time evolution of parameterized circuits. arXiv e-prints, page arXiv:2101.04579, Jan. 2021.

[3] F. Barratt, J. Dborin, M. Bal, V. Stojevic, F. Pollmann, and A. G. Green. Parallel quan-tum simulation of large systems on small nisq computers. npj Quantum Information, 7(1):79, May 2021.

[4] C. Cˆırstoiu, Z. Holmes, J. Iosue, L. Cincio, P. J. Coles, and A. Sornborger. Variational fast for-warding for quantum simulation beyond the co-herence time. npj Quantum Information, 6(1), Sep 2020.

[5] J. Eisert, M. Friesdorf, and C. Gogolin. Quantum many-body systems out of equilibrium. Nature Physics, 11(2):124–130, Feb 2015.

[6] M. Heyl, P. Hauke, and P. Zoller. Quantum local-ization bounds trotter errors in digital quantum simulation. Science Advances, 5(4), 2019. [7] B. D. M. Jones, D. R. White, G. O. O’Brien,

J. A. Clark, and E. T. Campbell. Optimising trotter-suzuki decompositions for quantum sim-ulation using evolutionary strategies. In Proceed-ings of the Genetic and Evolutionary Computa-tion Conference, GECCO ’19, page 1223–1231, New York, NY, USA, 2019. Association for Com-puting Machinery.

[8] D. P. Kingma and J. Ba. Adam: A method for stochastic optimization. CoRR, abs/1412.6980, 2015.

[9] I. D. Kivlichan, C. Gidney, D. W. Berry, N. Wiebe, J. McClean, W. Sun, Z. Jiang, N. Ru-bin, A. Fowler, A. Aspuru-Guzik, H. Neven, and R. Babbush. Improved Fault-Tolerant Quantum Simulation of Condensed-Phase Correlated Elec-trons via Trotterization. Quantum, 4:296, July 2020.

[10] M. Kjaergaard, M. E. Schwartz, J. Braum¨uller, P. Krantz, J. I.-J. Wang, S. Gustavsson, and W. D. Oliver. Superconducting qubits: Current state of play. Annual Review of Condensed Mat-ter Physics, 11(1):369–395, 2020.

[11] Y. Li and S. C. Benjamin. Efficient varia-tional quantum simulator incorporating active error minimization. Phys. Rev. X, 7:021050, Jun 2017.

[12] J. R. McClean, J. Romero, R. Babbush, and A. Aspuru-Guzik. The theory of variational hy-brid quantum-classical algorithms. New Journal of Physics, 18(2):023023, Feb 2016.

[13] J. Preskill. Quantum Computing in the NISQ era and beyond. Quantum, 2:79, Aug. 2018. [14] A. G. Rattew, S. Hu, M. Pistoia, R. Chen, and

S. Wood. A domain-agnostic, noise-resistant, hardware-efficient evolutionary variational quan-tum eigensolver, 2020.

[15] A. Smith, M. S. Kim, F. Pollmann, and J. Knolle. Simulating quantum many-body dynamics on a current digital quantum computer. npj Quantum Information, 5(1), Nov 2019.

[16] M. Suzuki. Decomposition formulas of expo-nential operators and lie expoexpo-nentials with some applications to quantum mechanics and statis-tical physics. Journal of Mathemastatis-tical Physics, 26(4):601–612, 1985.

[17] M. Suzuki. General theory of fractal path in-tegrals with applications to many-body theories and statistical physics. Journal of Mathematical Physics, 32(2):400–407, 1991.

[18] S. Trotzky, Y.-A. Chen, A. Flesch, I. P. Mc-Culloch, U. Schollw¨ock, J. Eisert, and I. Bloch. Probing the relaxation towards equilibrium in an isolated strongly correlated one-dimensional bose gas. Nature Physics, 8(4):325–330, Feb 2012.

A

Second-Order Suzuki

Decomposi-tion

An approximation which is better than Eq. (2) can be achieved by only reordering the operators in a sym-metric way (cf. Eq. (3)). We want to discuss two different approaches to a second order Suzuki approx-imation of the Transverse Field Ising model and its optimization.

If we modify the Trotter-Suzuki ansatz as a single time step unitary to the symmetrized ordering (Eq. (3), we can get rid of the leading order of approxi-mation error and are left with an error O(_mt33). The

Figure 9: Normalized cost values for up to 100 repetitions of the time evolution under a Transverse Field Ising Hamiltonian

with Jz = 0.25, hx = 1, t = 0.3 of two copies of a 6-qubit

system. The difference to Fig. 4is the symmetric ordering

of the operators in the ans¨atze. As a consequence, both,

the Trotter-Suzuki and the variational approximation enjoy

further improvement compared to Fig. 4.

then becomes U_varsym(~λ, ~ω) = m Y j=1 " _` Y i=1 i odd

e−iλi,j2 ZiZi+1 `

Y

i=1 i even

e−iλi,j2 ZiZi+1

× ` Y i=1 e−iωi,jXi ` Y i=1 i even

e−iλi,j2 ZiZi+1

×

`

Y

i=1 i odd

e−iλi,j2 ZiZi+1

#

. (20)

Note that, in general, the symmetric ordering raises the gate count of the final circuit by factor 1 < f < 2 dependent on the number of different layers of gates. For three layers A, B, C and m = 3, a single time step has the form

( √ A √ BC √ B √ A)m = √ A √ BC √ BA √ BC √ BA √ BC √ B √ A, (21) where we could combine √A√A = A except at the

boundary but we need double the gate count for layer

B. This can be extended for repetitive applications

of the circuit in order to simulate larger times. In our specific case, we do not raise the gate count significantly when going over to the symmetric opera-tor ordering. Since ZiZi+1commute for even and odd

i, the single time step circuit has the form

( √ A √ BC √ B √ A)m = √ A √ BCBACBAC √ B √ A. (22)

Figure 10: Error of the dynamics of the two-point spin cor-relation of a random input state for the system considered

in Fig. 9. The variational algorithm competes very well

with m = 6 Trotter-Suzuki decomposition also for large time steps. The symmetric ordering in the Trotter-Suzuki ansatz eliminated the oscillatory behavior which was observable in

the first order Suzuki variant (cf. Fig. 5).

Eq. (22) shows that for many repetitions of the cir-cuit, the additional gates required at the boundary of the circuit become negligible.

We tested this variation of the optimization scheme with the same parameters as in section4.1. The sym-metric ordering equivalent of Figs. 4and5are plotted in Figs. 9and10. We observe an overall improvement of cost value for Trotter-Suzuki and the variational ansatz of an order of magnitude. Also, the change of the leading error for Trotter-Suzuki has been made visible. In Fig. 4, the two-site correlation showed an oscillation with a period of Jzhxt2≈ ₄₀1 which is gone

in Fig. 10where only third order errors survive. A more elegant way making use of a second order Suzuki decomposition for larger total simulation times can be achieved by reverting the operator ordering in every second time step. Every odd circuit repetition will then be of the form of Eq. (14), whereas every even circuit repetition has the reversed form

Uvar(~λ, ~ω) = m Y j=1 " ` Y i=1 e−iωi,jXi ` Y i=1 i even

e−iλi,jZiZi+1

×

`

Y

i=1 i odd

e−iλi,jZiZi+1

#

. (23)

For λi,j = _mtJz, ωi,j = _mthx we end up having a

second order Suzuki ansatz with double the time 2t. Note that inverting the ordering in every second step comes at no extra cost. While a larger time lowers the accuracy of a Trotter-Suzuki approximation, the symmetric ordering raises it. We hence expect this method to be fruitful at a low number of circuit repe-titions, but to get worse than the normal ordering for large times, when the cubic error grows faster.

Figure 11: Normalized cost values for up to 100 repetitions of the time evolution under a Transverse Field Ising

Hamil-tonian with Jz = 0.25, hx = 1, t = 0.3 of two copies of a

6-qubit system. The difference to Fig. 4 is the inversion

of the ordering of the operators in every second time step.

The Trotter-Suzuki ans¨atze are equivalent to a second order

Suzuki decomposition with τ = 0.6 while a single time step unitary is a first order Suzuki decomposition with τ = 0.3.

Ising model. While the variational approach seems to be stable under this variation, the Trotter-Suzuki ansatz shows changes similar to the symmetric order-ing discussed before. In Fig. 11, we study the cost value over 100 time steps, again. For a few time steps we observe an improvement compared to first order Suzuki (cf. Fig. 4) which gets lost for higher circuit repetitions. The two-site correlation (Fig. 12), on the other hand, admits a higher error compared to Fig.

5. The absence of the oscillation in that error can be traced back again to the fact that we consider a second order Suzuki approximation, but now with a time step of t = 0.6 instead of t = 0.3.

In all additional cases discussed in this section, the variational ansatz outperforms the Trotter-Suzuki ansatz. In some situations, it is hence advantageous to go over to a second order Suzuki ansatz. When there are no three mutually non-commuting layers in the considered circuit, the symmetric ordering raises the overall precision without significantly raising the gate count.

The second approach to a symmetric ordering does not require a raise of the gate count at all, but natu-rally comes with a doubling of the time step t 7→ 2t. Also, the variational approach will not be improved since the parameters will already be optimized to compensate for the leading order of Trotter-Suzuki error. As a consequence, inverting the order of every second time step unitary will only be advantageous for the Trotter-Suzuki ansatz at a small number of

Figure 12: Error of the dynamics of the two-point spin cor-relation of a random input state for the system considered in Fig. 11. Also here, the symmetric ordering in the Trotter-Suzuki ansatz eliminated the oscillatory behavior which was

observable in the first order Suzuki variant (cf. Fig. 5).

However, since the simulation time is doubled in second or-der Suzuki decomposition from the inversion of the oror-dering in every second time step unitary with τ = 0.3.