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Differentiable matrix product states for simulating variational quantum computational chemistry

Chu Guo, Yi Fan, Zhiqian Xu, Honghui Shang

arXiv:2211.07983v4quant-ph

TL;DR

Classical VQE simulation faces state-vector memory limits, while full configuration interaction scales exponentially. This paper combines an MPS circuit simulator with automatic differentiation for gradients, and reports chemical accuracy for several small molecules alongside simulations of larger systems up to 40 qubits.

  • Problem

    State-vector VQE simulation is constrained by exponential memory, while exact full configuration interaction scales exponentially with system size.

    Source: §1 · p. 1

  • Method

    The paper integrates an MPS-based quantum circuit simulator with reverse-mode automatic differentiation to compute VQE gradients.

    Source: §4 · p. 5, §6 · p. 10

  • Results

    Chemical accuracy is reached for HF, HCl, LiH, and H2O across bond lengths from 0.5 to 2.0 Å, and larger-molecule applications extend to 40 qubits.

    Source: §1 · p. 2, §5 · p. 8

  • Takeaways & Limitations

    The authors present differentiable MPS as a promising testing ground for variational quantum algorithms, with complexity independent of the number of variational parameters.

    Source: §6 · p. 10

    Convergence of large molecules toward chemical accuracy using more involved variational circuit ansatzes is beyond this work’s scope.

    Source: §5 · p. 10

Abstract

from arXiv · show

Quantum Computing is believed to be the ultimate solution for quantum chemistry problems. Before the advent of large-scale, fully fault-tolerant quantum computers, the variational quantum eigensolver~(VQE) is a promising heuristic quantum algorithm to solve real world quantum chemistry problems on near-term noisy quantum computers. Here we propose a highly parallelizable classical simulator for VQE based on the matrix product state representation of quantum state, which significantly extend the simulation range of the existing simulators. Our simulator seamlessly integrates the quantum circuit evolution into the classical auto-differentiation framework, thus the gradients could be computed efficiently similar to the classical deep neural network, with a scaling that is independent of the number of variational parameters. As applications, we use our simulator to study commonly used small molecules such as HF, HCl, LiH and H$_2$O, as well as larger molecules CO$_2$, BeH$_2$ and H$_4$ with up to $40$ qubits. The favorable scaling of our simulator against the number of qubits and the number of parameters could make it an ideal testing ground for near-term quantum algorithms and a perfect benchmarking baseline for oncoming large scale VQE experiments on noisy quantum computers.

1 Introduction

Exact full configuration interaction scales exponentially and is limited to about 20 orbitals, while state-vector VQE simulation is constrained by memory. The paper proposes an MPS simulator with automatic-differentiation gradients and distributed parallelization, and applies it to molecules with up to 40 qubits.

1 Introduction

  • Exact full configuration interaction scales exponentially with system size and is strictly limited to within 20 orbitals on classical computers.Source: §1 · p. 1
  • State-vector VQE simulators require storing 2^N complex numbers, making simulations beyond 50 qubits extremely difficult even on powerful supercomputers.Most reported classical VQE simulations are limited to within 30 qubits.Source: §1 · p. 1
  • The paper positions classical simulation as a reference for benchmarking noisy quantum computations and for studying circuit ansatzes without a quantum computer.Source: §1 · p. 1
  • MPS simulation is applied to larger molecules including CO2, BeH2, and H4 with up to 40 qubits, beyond the range where state-vector simulations are mostly limited.For HF, HCl, LiH, and H2O, the simulator reaches accuracies comparable to existing approaches.Source: §1 · p. 2
  • The proposed MPS simulator integrates circuit evolution with automatic differentiation and distributed parallelization, making gradient complexity independent of variational-parameter count.Source: §1 · p. 2

2 VQE for quantum chemistry

The section introduces VQE as a hybrid procedure that optimizes a circuit-encoded ground state and describes the chemistry ansatz and its circuit construction. It also notes that shallow circuits can represent some states that are difficult to handle with classical tensor-network methods.

2 VQE for quantum chemistry

  • VQE prepares a circuit-encoded ground state and minimizes the Hamiltonian expectation over variational parameters using hybrid quantum-classical optimization.Source: §2 · p. 2
  • Shallow quantum circuits can generate any MPS and some PEPS instances, which may represent certain Hamiltonian ground states better than commonly used classical ansatzes.The passage notes that manipulating PEPS classically can be very hard.Source: §2 · p. 2
  • The work uses the physically motivated UCC ansatz, typically referenced to the Hartree-Fock state, and allows other ansatzes in principle.For UCCSD, the cluster operator is truncated to single and double excitations.Source: §2 · p. 2
  • Trotter-Suzuki decomposition converts the UCC operator exponential into successive V-shaped circuit blocks, with finite-order depth approximately O(kN^4).Here k is the Trotterization order and N is the number of qubits.Source: §2 · p. 3

3 MPS based quantum circuit simulator

The simulator represents quantum states as MPS and implements state preparation, gate updates, and expectation-value calculations. Its gate procedures preserve a right-canonical form and use Schmidt-value truncation to control the representation size.

3 MPS based quantum circuit simulator

  • A general state’s 2^N amplitudes can be represented as an MPS chain of rank-3 tensors, whose virtual-index size sets the bond dimension and computational cost.An MPS represents any state exactly if its bond dimension grows exponentially; near-constant bond dimension gives efficient representation as N grows.Source: §3 · p. 3
  • The simulator’s three required functions are preparing a separable initial state, applying gates while updating the MPS, and computing Hamiltonian-term expectations.The Hartree-Fock reference is separable and can be represented as an MPS with D = 1.Source: §3 · p. 3
  • Single-qubit and nearest-neighbour two-qubit gates update the MPS while preserving its right-canonical form; two-site updates use SVD and truncate small Schmidt values or retain at most D values.Non-nearest-neighbour gates can be decomposed into nearest-neighbour gates using SWAPs, and gate fusion can improve efficiency.Source: §3 · pp. 3–4
  • A Pauli-string expectation is computed by contracting a one-dimensional tensor network with complexity O(ND^3).Source: §3 · p. 4

4 Back propagation algorithm for MPS

The section adapts reverse-mode automatic differentiation to MPS VQE, using reversible circuit evolution and grouped Hamiltonian terms to control memory and parallelize gradient evaluation. Its accuracy-cost balance depends on fixed-bond-dimension truncation and the Hamiltonian split.

4 Back propagation algorithm for MPS

  • Reverse-mode automatic differentiation computes gradients at roughly the cost of evaluating the loss, while reversible gates enable a memory-efficient scheme using only two quantum-state copies.The two-copy scheme avoids storing an intermediate state for every gate operation.Source: §4 · pp. 4–5
  • The method treats circuit evolution and final-state expectation evaluation as elementary functions, then propagates adjoints backward to obtain the parameter gradients.The resulting gradient evaluation can use the memory-efficient procedure without additional state copies beyond its two input states.Source: §4 · p. 5
  • Summing many Pauli-string-applied MPS can inflate bond dimension, so the method divides the Hamiltonian into groups and computes each group's gradients at fixed bond dimension.Because back-propagation truncation is used, the group count and assignment of Pauli strings can affect final gradient accuracy.Source: §4 · p. 6
  • Hamiltonian terms are distributed across MPI processes, with no further communication required because expectation and gradient calculations depend only on each process's local Hamiltonian.The illustrated H2 Hamiltonian contains 15 Pauli strings.Source: Figure 2 · p. 6, §4 · p. 6
  • Against parameter-shift's 2M loss evaluations, grouped back propagation is more efficient when the number of Hamiltonian groups m is much smaller than the parameter count M.Gradients for each Hamiltonian group can be computed almost perfectly in parallel.Source: §4 · pp. 4, 6
  • A local site-tensor gradient approach could optimize within fixed bond dimension, but it requires saving intermediate calculations and may involve unstable SVD back propagation.In the global-state approach, gradients lose accuracy if intermediate states cannot be represented at the chosen bond dimension.Source: §4 · p. 7

5 Results

The simulator’s gradients converge toward the parameter-shift results as bond dimension grows, while its speedup scales approximately linearly with the number of parametric gates. Tests range from small-molecule potential energy curves to 40-qubit simulations, where the authors identify entanglement entropy and ansatz expressiveness as important boundaries.

5 Results

  • Gradient error against parameter shift decreased as bond dimension D grew, while differentiable-MPS speedup over parameter shift scaled approximately linearly with the number of parametric gates M.Figure 3 compares LiH, H2O, and O2; its speedup panel uses D = 128 and Nh/m = 8.Source: §5 · p. 7, Figure 3 · p. 7
  • For H2O, mean relative gradient error increased with the number of Pauli strings per group and stabilized around 10^-3 when Nh/m ≥ 8 (m ≤ 48).The calculation used the STO-3G basis set and 14 qubits.Source: §5 · p. 7, Figure 4 · p. 7
  • Gradient-based solvers converged much faster than BOBYQA for H2O and N2, with BFGS reaching the lowest energy in fewer than 50 epochs; BOBYQA could still fluctuate after 100 epochs.The comparison used Nh/m = 4.Source: §5 · p. 8, Figure 5 · p. 8
  • Across HF, HCl, LiH, and H2O bond lengths from 0.5 to 2.0 Å, the potential-energy curves reached chemical accuracy; BFGS generally gave the best accuracy.Chemical accuracy is defined here as an error within 1.6×10^-3 Hartree or 1 kcal/mol.Source: §5 · p. 8, Figure 6 · p. 9
  • The method’s main scaling boundary is entanglement entropy, which determines the bond dimension needed for a target precision, rather than qubit count, circuit depth, or parameter count.The authors also note that the simulations up to this point had used no more than 20 qubits.Source: §5 · p. 8
  • For the 30-qubit CO2 reaction calculation, the energy barrier was −21.0 kcal/mol versus −25.5 kcal/mol for FCI, a 4.5 kcal/mol error mainly attributed to the screened UCCSD operator pool.The authors note that UCCGSD or ADAPT-VQE could yield more accurate results.Source: §5 · p. 8
  • BeH2 and H4 energies quickly stabilized about 10^-2 Hartree above FCI; the authors attribute the gap likely to limited ansatz expressiveness and leave chemical-accuracy convergence beyond scope.The simulations used 36 and 40 qubits, respectively.Source: §5 · pp. 8–10

6 Conclusions

The paper introduces differentiable MPS simulation with memory-efficient backpropagation, whose complexity is independent of the number of variational parameters. Its accuracy depends on MPS bond dimension, so simulations should be checked across bond dimensions.

6 Conclusions

  • Differentiable MPS integrates quantum-circuit simulation with automatic differentiation using memory-efficient backpropagation that stores only two quantum-state copies at minimum.The method is described as less sensitive to qubit count, with gradient-computation complexity independent of the number of variational parameters.Source: §6 · p. 10
  • MPS bond dimension can significantly affect simulation accuracy according to the entanglement generated in the quantum state, and calculated gradients are subject to the same constraint.The paper recommends checking convergence across different bond dimensions.Source: §6 · p. 10
  • Projected entangled pair states and tree tensor networks are identified as possible alternatives to explore for quantum-circuit simulation.Source: §6 · p. 10

A Back propagation rule for loss function with complex numbers

The appendix describes backpropagation for real-valued losses of complex variables by viewing the loss as a function of real and imaginary components. It extends the chain rule to compositions of complex functions and relates these rules to the main-text derivation.

A Back propagation rule for loss function with complex numbers

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