Parallel Quantum Simulation of Large Systems on Small Quantum Computers

I. INTRODUCTION

• Tensor networks use entanglement as a resource for accurate approximate descriptions of large quantum systems and for concentrating computation in useful regions of Hilbert space.Finite-depth NISQ circuits face gate-fidelity and entanglement-resource limits.Source: §I · p. 1
• The paper demonstrates parallel quantum simulation of systems larger than available NISQ machines, with Cirq and Qiskit code translating iMPS algorithms for infinite, translationally invariant spin chains.The circuits represent, optimise, and evolve these states.Source: §I · pp. 1–2

A. Parallel Quantum Simulation Across Weakly-Entangled Cuts

• Across a weakly entangled cut, Schmidt decomposition lets each partition’s influence on the other be represented by an effective state on log2 D spins.The environment unitary encodes the Schmidt coefficients and is determined by solving the quantum problem on the opposite side.Source: A. Parallel Quantum Simulation Across… · p. 2
• The two partitions can be simulated on separate circuits, with measurements used to update their environment unitaries iteratively to consistency.Full state tomography scales as O(poly(D)); more sophisticated methods can reduce this to O(log(D)).Source: A. Parallel Quantum Simulation Across… · p. 2

Representing the state

• A translationally invariant spin-1/2 MPS of bond order D = 2^N is represented by an infinite circuit, while local expectations are evaluated with a finite circuit and a self-consistent environment unitary V.V encodes an effective state over N = log2 D spins; unitarity makes contraction to the left of the observable trivial.Source: B. Parallel Simulation with Quantum… · p. 2, Figure 2 · p. 3

Optimising the state

• Ground-state energy density is minimised by measuring local Hamiltonian Pauli strings and updating the circuit parameters together with V.Interleaving the environment updates keeps the optimisation over valid translationally invariant states.Source: B. Parallel Simulation with Quantum… · p. 3

Evolving the state

• Time evolution updates U(t) to U(t + dt) through a circuit whose first variation yields the iMPS time-dependent variational principle.The update also requires nested iterations for the mixed environment tensors L and R.Source: B. Parallel Simulation with Quantum… · p. 3, Figure 3 · p. 4

Quantum Advantage

• Shallow circuits can reduce contraction from O(D^3) classically to O(log D) on a quantum circuit for a subset of non-trivial unitaries with entanglement S.The cited shallow-circuit subset is reported to have high fidelity with states from Hamiltonian evolution; the environment circuits also require shallow representations.Source: B. Parallel Simulation with Quantum… · p. 3
• Shallow environment approximations can be improved exponentially for linear cost in qubits and circuit depth by repeatedly applying the transfer matrix.For energy optimisation, the reduced density matrix can be diagonalised; time evolution needs a richer shallow parameterisation because L and R need not be simultaneously diagonalizable.Source: B. Parallel Simulation with Quantum… · p. 4

• With full unitary representations and no gate errors, the code reproduces optimum iMPS and TDVP evolution; factorization errors improve with ansatz depth, while accurate evolution needs deeper, wider circuits.Source: C. Numerical Results · p. 4
• Shallow state-unitary factorizations capture dynamical phase transitions in Loschmidt echoes, but better agreement requires more accurate circuits.The Ising-model time-evolution circuits reproduce the TDVP equations with exact unitary representations.Source: C. Numerical Results · pp. 4–5
• The Fig. 4a factorization agrees well with exact bond-dimension-2 MPS optimization at low circuit depth; increasing parametrization depth improves energy toward conventional MPS results.At λ = 0, the product-state optimum is captured with p = 1.Source: C. Numerical Results · pp. 4–5, Figure 4 · p. 5
• For the PXP model, quantum TDVP recreates Poincare maps from low-depth states, with larger structures remaining visible but distorted by integration and stochastic-optimization errors.The maps use two-site translationally invariant MPS states at bond order D = 2.Source: C. Numerical Results · p. 5, Figure 5 · p. 5

III. DISCUSSION

• The approach translates tensor-network algorithms into quantum circuits for parallel simulation of large one-dimensional translationally invariant spin chains on small NISQ computers.The authors identify shallow-circuit expressiveness and finite gate fidelity as questions requiring further systematic study.Source: §III · p. 6
• The algorithms are described as readily extensible to inhomogeneous one-dimensional systems and higher dimensions, including through existing methods for wrapping one-dimensional states.The discussion also points to isometric PEPS and unitary-based MERA as possible circuit realizations.Source: §III · p. 6
• Potential quantum advantage for PEPS is conditional on an efficiently contractible, physically relevant subset with an isometric representation that permits exact, quasi-local contraction.General PEPS contraction is described as #P hard.Source: §III · p. 6
• Extracting circuit-state tensor elements requires tomography that is exponentially slow in the number of measured spins, potentially bottlenecking hybrid algorithms.The authors say distributing entanglement resources appropriately can mitigate some costs.Source: §III · p. 6
• The work frames tensor-network simulation as a route to quantum advantage while also offering a complementary perspective on classical algorithms.Source: §III · p. 6

A. Quantum Matrix Product States

• The MPS-to-circuit mapping encodes the left canonical form and tangent-space structure, yielding a compact TDVP implementation that avoids recalculating the tangent space at each step.Classical MPS algorithms restore canonical form using singular value decompositions; the unitary realization does not require that manipulation.Source: A. Quantum Matrix Product States · p. 6

B. The Quantum Time-dependent Variational Principle

• Differentiating the circuit overlap with respect to X recovers the TDVP equations, with X rescaled relative to Ref. [12] by the square root of the environment tensor.The equivalence is shown by expanding the overlap to quadratic order in X and bilinear order in X and dt before differentiation.Source: B. The Quantum Time-dependent… · pp. 6–7

C. Optimising Quantum Circuits

• Stochastic optimization targets observable expectations and fixed-point equations for the environment and mixed environments.The fixed-point equations are identities between density matrices and are optimized using a swap-test variant.Source: C. Optimising Quantum Circuits · p. 7
• Rotosolve exploits sinusoidal parameter dependence to reach a parameter’s local optimum after three measurements.Extensions handle variations where several circuit elements share a parameter.Source: C. Optimising Quantum Circuits · p. 7
• Swap-test implementations for the environment and mixed environments are detailed in the supplementary materials.These correspond to the environment in Fig. 2b and mixed environments in Figs. 3b and 3c.Source: C. Optimising Quantum Circuits · p. 7

VI. CODE AVAILABILITY STATEMENT

• Both Cirq and Qiskit code with the same functionality are available, and the reported simulations used Cirq.The code is available from the stated project repository.Source: §VI · p. 7

VIII. AUTHOR CONTRIBUTIONS

• AGG conceived the project; FB, JD, and AGG developed the detailed mapping, wrote code, and ran simulations, while FP contributed to code development.FB, JD, and AGG wrote the manuscript.Source: §VIII · p. 7

Supplementary Materials: Parallel Quantum Simulation of Large Systems on Small

• The supplementary material covers practical circuit realisation, ansätze, optimisation, higher bond orders, and effects of shallow circuits and finite gate fidelity.It proceeds from MPS-to-circuit ideas to practical realisation and additional data on circuit imperfections.Source: Supplementary Materials: Parallel… · p. 10

1. Matrix Product States

• Contracting the chain on either side of a site defines left and right environments, allowing local expectations to be expressed using those environments.The contractions otherwise extend along the whole chain.Source: §1 · p. 10
• Sequential left-to-right singular-value decomposition puts MPS tensors in left-orthogonal form, making the left environment the identity and providing the basis for the circuit mapping.MPS tensor indices have bond dimensions D_i; in practice these are usually truncated.Source: §1 · p. 10

2. Translation to Quantum Circuit

• The right environment r is decomposed into eigenvectors and eigenvalues to construct V; retaining its k largest eigenpairs gives the best rank-k approximation in Frobenius norm.The construction embeds X, reshaped as a vector, in V’s first column because V acts on |0⟩^⊗N; the remaining columns are chosen orthonormally.Source: §2 · p. 11
• The right environment can also be embedded in a diagonal block of R, with the unused elements completed by a singular-value decomposition.Source: §2 · p. 11
• Circuit environments are built iteratively from site to site; translational invariance makes V and R site-independent and reduces the construction to a fixed-point equation.The fixed-point equation corresponds to the one shown in Fig. 2c.Source: §2 · p. 11

3. Time-evolving Quantum Circuit MPS

• Optimizing the fidelity to the time-evolved state sets the updated circuit unitaries to encode the state at time t + dt.The unitaries before the update encode the state at time t, and the overlap uses a second-order Trotter expansion.Source: §3 · p. 11
• Splitting the circuit along cuts permits implementation on a quantum computer much smaller than the system being simulated.The cut yields mixed-environment circuits compressed into tensors Rα and Lα, which can be determined iteratively.Source: §3 · pp. 11–12
• Optimizing the evolution overlap alongside the environment equations yields evolution equivalent to TDVP over the variational parameters of U.Setting the derivative of the overlap objective with respect to U′ to zero recovers the usual TDVP equations; the unitary encoding also includes the tangent space.Source: §3 · p. 12
• For translationally invariant systems, the environment unitaries become site-independent and their updates reduce to fixed-point equations for transfer-matrix eigenvectors.Source: §3 · p. 12
• A first-order Trotterization is used to simplify implementation on NISQ devices, and translational invariance appears to reduce the resulting errors somewhat.Source: §3 · p. 12
• The main-body simulations use bond order D = 2 for compact circuits, although the Cirq and Qiskit code supports arbitrary bond order.Restricting to D = 2 recovers the circuits in Fig. 3 of the main text.Source: §3 · p. 12

1. Finding the Environment

• A plain SWAP-test overlap can favor purity rather than matching when the reduced density matrices are mixed.The trace-distance-related quantity is minimized at ρ = σ.Source: §1 · p. 13

2. Finding the Overlap

• The infinite-chain overlap is defined by the largest eigenvalue of the mixed transfer matrix, found by variationally minimizing the distance between its action on a vector and a candidate eigenvalue times that vector.Because the transfer matrix is non-Hermitian, the Rayleigh-Ritz principle cannot be used.Source: §2 · p. 13
• For similar states, the search jointly optimizes η and r from an initial η near 1 and restarts if the final η is not near 1.The method uses the expectation that η deviates from 1 only by O(δt).Source: §2 · p. 13
• Circuit evaluation measures the all-zero bit-string probability and takes its square root; the first two objective terms are directly implemented, while the last two have an undetermined phase.The latter terms sum to the largest eigenvalue at the minimum, real to O(dt^2).Source: §2 · p. 13
• The method determines the overlap with accuracy often much greater than O(dt^2), and supplementary data demonstrate its accuracy and reliability.The variational largest-eigenvalue search rarely requires restarts, and any needed restarts typically are few.Source: §2 · p. 13

State unitary

• The generic state unitary ansatz is a workable tradeoff between expressibility and depth, while different unitary factorizations can reflect their distinct structures.At low bond dimension, full parametrizations of important matrix spaces are possible.Source: §3 · p. 14

Symmetry-adapted ansätze

• A real state-tensor ansatz reaches the classical D=2 optimum at depth 4 for the transverse-field Ising model.The supplementary figure compares quantum variational optimization at different ansatz depths with classical iDMRG at D=2.Source: §3 · p. 14, Figure 4 · p. 15

Representing the environment

• At D=2, the positive, trace-one right environment is represented by embedding its Cholesky decomposition in a unitary V.The environment parametrization allows quantities such as Schmidt coefficients to be calculated classically.Source: §3 · p. 14
• For D>2, the environment ansatz uses a tensor-product family of positive diagonal matrices rather than spanning all possible diagonal matrices.The generalization is reported effective in practice, but theoretical guarantees remain open.Source: §3 · pp. 14–16

Alternative environment representation

• The alternative TDVP environment representation trades circuit depth for qubits, exposes time-reversal symmetry, and permits variational exploration of non-Hermitian tensors.Its environment eigenvalues are the squared Schmidt coefficients across each bond.Source: §3 · p. 16

4. Optimization Methods

• When one parameter controls two gates, its expectation value is a sum of sinusoids with single and doubled frequencies, requiring six measurements and classical minimization.The minimum is not obtained analytically as in original Rotosolve.Source: §4 · pp. 16–17
• The modified algorithm suffers no significant loss of performance over Rotosolve and requires only 3 more measurements per iteration.The modification addresses circuits in which a parameter controls more than one gate.Source: §4 · pp. 16–17

II. SUPPLEMENTARY DISCUSSION

• Time evolution is more susceptible to gate errors than state representation, restricting reliable evolution to lower bond order even when higher-bond-order states can be represented.Increasing noise degrades but does not eliminate the dynamical phase-transition peaks.Source: §II · p. 17, Figure 7 · p. 17
• The framework trades noise against performance through circuit depth or a shallow, higher-bond-order ansatz that requires more qubits; near-term-chip performance remains uncertain.Supplementary Fig. 7 also notes a tradeoff between circuit depth and sensitivity to noise.Source: §II · p. 17, Figure 7 · p. 17
• The Poincaré map is constructed by interpolating parameter trajectories, finding a crossing time, and evaluating the other parameters at that time.The reported crossing values are obtained by polynomial interpolation of discretized circuit output.Source: §II · p. 17