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Pre-optimizing variational quantum eigensolvers with tensor networks

Abid Khan, Bryan K. Clark, Norm M. Tubman

arXiv:2310.12965v1quant-phcond-mat.str-el

TL;DR

Random initialization can make VQE optimization difficult, while classical circuit simulation offers a route to pre-optimization. The paper introduces VTNE, which approximates VQE circuits with limited-bond-dimension MPSs, and reports improved starting configurations and fewer gradient evaluations, with 1D errors remaining low across system sizes.

  • Problem

    Randomly parameterized PQCs can encounter barren plateaus and local minima, making optimization challenging.

    Source: §I · p. 1

  • Method

    VTNE classically pre-optimizes VQE parameters by representing the circuit as an MPS with fixed bond dimension χ.

    Source: §I · p. 2, §II · p. 2, C. MPS from Quantum Circuit · p. 3

  • Results

    For 1D systems up to 32 qubits, χ = 16 gives ≲3% relative energy error, while MPS-preoptimized starts save about 400 gradient calls at χ = 16 and 1,000 at χ = 32.

    Source: §III · pp. 6–7

  • Takeaways & Limitations

    When circuit depth is limiting, VTNE followed by VQE is a competitive and efficient approach for achieving low-energy states.

    Source: §IV · p. 7

    VTNE constrains the circuit approximation to a fixed bond dimension χ significantly smaller than the maximum χmax = 2^nq/2.

    Source: §II · p. 2

Abstract

from arXiv · show

The variational quantum eigensolver (VQE) is a promising algorithm for demonstrating quantum advantage in the noisy intermediate-scale quantum (NISQ) era. However, optimizing VQE from random initial starting parameters is challenging due to a variety of issues including barren plateaus, optimization in the presence of noise, and slow convergence. While simulating quantum circuits classically is generically difficult, classical computing methods have been developed extensively, and powerful tools now exist to approximately simulate quantum circuits. This opens up various strategies that limit the amount of optimization that needs to be performed on quantum hardware. Here we present and benchmark an approach where we find good starting parameters for parameterized quantum circuits by classically simulating VQE by approximating the parameterized quantum circuit (PQC) as a matrix product state (MPS) with a limited bond dimension. Calling this approach the variational tensor network eigensolver (VTNE), we apply it to the 1D and 2D Fermi-Hubbard model with system sizes that use up to 32 qubits. We find that in 1D, VTNE can find parameters for PQC whose energy error is within 0.5% relative to the ground state. In 2D, the parameters that VTNE finds have significantly lower energy than their starting configurations, and we show that starting VQE from these parameters requires non-trivially fewer operations to come down to a given energy. The higher the bond dimension we use in VTNE, the less work needs to be done in VQE. By generating classically optimized parameters as the initialization for the quantum circuit one can alleviate many of the challenges that plague VQE on quantum computers.

I. INTRODUCTION

VQE suits NISQ devices, but random circuit initialization can make optimization difficult. The paper presents VTNE as a way to use classical tensor-network simulation to pre-optimize VQE parameters before quantum hardware is used.

I. INTRODUCTION

  • VQE is suited to NISQ devices because it can use shallower circuits than more exact algorithms and its variational gate parameterization can provide some error mitigation.The passage cites applications in chemistry, materials science, and machine learning.Source: §I · p. 1
  • Randomly parameterized PQCs face barren plateaus and local minima, complicating efforts to realize advantages over classical methods.VQE circuits for molecular ground states have also used Hartree-Fock or coupled-cluster-based initializations.Source: §I · p. 1
  • Tensor-network pre-optimization is one way to explore classical pre-optimization strategies amid increasingly sophisticated classical circuit simulation.The authors note that tensor networks can use GPU and TPU accelerators.Source: §I · p. 1
  • Classical pre-optimization with tensor networks produces intermediate VQE parameters before optimization continues on quantum hardware.The paper names this approach the variational tensor network eigensolver (VTNE).Source: §I · pp. 1–2

II. METHODS

VTNE classically approximates a fixed VQE circuit as an MPS to estimate energy and its derivatives. A fixed bond dimension limits the representation of highly entangled circuit states and controls classical optimization cost.

II. METHODS

  • For each parameter set, the fixed hardware circuit is mapped to an approximate MPS used to calculate energy and its derivative.The MPS is constructed by translating each unitary gate into a rank-2d tensor, where d is the number of qubits it interfaces with, and contracting the network.Source: §II · p. 2
  • Each gate contraction caps the MPS bond dimension at χ.Figure 1 depicts the quantum circuit being contracted into an MPS.Source: Figure 1 · p. 2
  • The circuit is approximated with a fixed bond dimension χ significantly smaller than χmax = 2^nq/2, limiting representation of highly entangled states.The bond dimension controls the computational complexity of classical optimization.Source: §II · p. 2

A. Model

The paper studies one- and two-dimensional Fermi-Hubbard systems, using a half-filled regime with t = 1 and U = 2 and a Jordan-Wigner qubit encoding.

A. Model

  • The study considers the one- and two-dimensional Fermi-Hubbard model, whose regular structure and simple form suggest it may be easier to implement on NISQ devices.The authors anticipate that their high-level approaches may also apply to other condensed-matter systems.Source: A. Model · p. 2
  • The fermionic operators create and annihilate particles, while nearest-neighbor pairs define the hopping terms; t is hopping and U is the on-site potential.The model uses t = 1 and U = 2 throughout.Source: A. Model · p. 2
  • The simulations use the half-filled regime and encode the fermionic Hamiltonian as a qubit Hamiltonian with the Jordan-Wigner transformation.The passage identifies this as the well-known Jordan-Wigner encoding.Source: A. Model · p. 2

B. Ansatz

The ansatz preserves particle number and combines parameterized gates with fermionic swaps to implement hopping on 1D and 2D lattices. It also includes initial Rz gates that the authors found helped optimization.

B. Ansatz

  • The number-preserving ansatz is more general for the Fermi-Hubbard model than its associated Hamiltonian variational ansatz (HVA).It uses parameterized number-preserving gates and parameter-less fermionic swap gates.Source: B. Ansatz · p. 2
  • The circuit prepends Rz(θ) gates at each qubit, and testing found that this layer helped optimization.Source: B. Ansatz · p. 3
  • Each layer applies onsite interactions, then horizontal and vertical hopping sets, using fermionic swaps to bring same-spin neighboring wires together and restore their order.Figure 2 illustrates the layer for a 2×2 lattice.Source: Figure 2 · p. 3, B. Ansatz · p. 3

C. MPS from Quantum Circuit

The circuit is approximated as a bond-dimension-χ MPS so its energy can be optimized classically. Increasing χ brings the optimized energy closer to the exact energy.

C. MPS from Quantum Circuit

  • The circuit is approximated as a bond-dimension-χ MPS using TEBD, yielding a truncated representation for classical energy optimization.The starting state |ψ0⟩ is unparameterized.Source: C. MPS from Quantum Circuit · p. 3
  • The objective evaluates the MPS energy with H represented as a matrix product operator; k-local Hamiltonians generally have simple MPO implementations.Source: C. MPS from Quantum Circuit · p. 3
  • As χ increases, the optimized energy Eχ gets closer to Eexact; at χmax = 2^nq/2, Eexact(θ) = Eχmax(θ).Source: C. MPS from Quantum Circuit · p. 3

D. Optimization

The optimization begins with a non-interacting state and uses its parameters to initialize the interacting case. An approximate, memory-saving gradient scheme supports classical optimization, with truncation limiting its correspondence to the objective gradient.

D. Optimization

  • For U=0, separate number-preserving ansätze place spin-up on even sites and spin-down on odd sites, forming a checkerboard configuration for the subsequent U=2 optimization.Source: D. Optimization · pp. 3–4
  • The non-interacting optimization uses half the qubits and removes onsite gates, cross-flavor swap gates, and cross-flavor hopping terms, leaving less than half the parameters to optimize.Source: D. Optimization · p. 3
  • The minimization uses BFGS and terminates when function tolerance reaches 10^-7 or gradient norm reaches 10^-6.Source: D. Optimization · p. 3
  • The approximate gradient scheme uses less memory and only two circuit evaluations; it is exact at maximal bond dimension but, after truncation to χ, approximates the derivative of Eq. 6 rather than Eq. 5.Source: D. Optimization · p. 4

III. RESULTS

Across the Hubbard-model benchmarks, increasing MPS bond dimension improves energy accuracy, while classically optimized parameters reduce the VQE gradient calls needed to reach lower energies. The results are strongest in 1D; larger 2D systems remain more challenging.

Layer depth

  • For the 8 × 1 system, χ = 16 fully represents a one-layer ansatz, but additional layers require larger bond dimensions; χ = 16 and 32 do not fully represent even one layer of the 4 × 2 system.The layer-depth comparison also shows that χ = 32 ceases to represent the 8 × 1 circuit exactly at four layers.Source: §III · p. 4, Figure 4 · p. 5

Energy accuracy

  • With optimized parameters retained, evaluation at a larger bond dimension gives comparable or better energies; at full bond dimension, energies are equal or lower.The results distinguish the optimization bond dimension from the bond dimension used to evaluate the circuit.Source: §III · p. 5
  • For all 1D systems up to 32 qubits, bond dimension χ_b = 16 gives a relative energy error of ≲3%.The reported 1D errors depend only weakly on system size.Source: §III · p. 6
  • In 2D, relative energy errors tend to increase with lattice size, making accurate ground-state energies more challenging as entanglement complexity grows.The authors report a similar bond-dimension trend for classical VQE and DMRG.Source: §III · p. 6

VQE initialization

  • Classically optimizing beforehand with an MPS backend significantly reduces gradient evaluations needed to reach lower energies, saving about 400 calls at bond dimension 16 and about 1,000 at bond dimension 32.Source: §III · p. 7

IV. DISCUSSION

VTNE optimizes parameters for a chosen circuit, which can be shallow or hardware-tailored, rather than converting a best-bond-dimension MPS directly into a circuit. When circuit depth is limited, that distinction makes VTNE followed by VQE a competitive route to low-energy states.

IV. DISCUSSION

  • VTNE finds parameters for a chosen, potentially shallow or hardware-commensurate circuit, while direct MPS-to-circuit approaches require deeper circuits and offer less architectural freedom.The discussion notes that a χ = 64 MPS could require up to 7,660 CNOT gates per site when implemented as a circuit.Source: §IV · p. 7
  • Direct MPS-to-circuit construction has a depth-dependent bond-dimension constraint, χ ≲ O(D/N), where D is circuit depth and N is system size.The discussion contrasts this constraint with VTNE's parameter optimization for a selected circuit architecture.Source: §IV · p. 7
  • At fixed circuit depth, DMRG-generated MPSs are limited to system-size-dependent bond dimensions, resulting in higher-energy states than VTNE as system size increases.The paper identifies VTNE followed by VQE as competitive and efficient when circuit depth is limiting.Source: §IV · p. 7

V. CONCLUSIONS

The conclusions report that VTNE can perform most or all optimization classically for the studied 1D systems, while for 2D systems it supplies VQE with an initial parameter set. Figure 7 compares optimization steps for different initialization strategies on the 4×2 Hubbard model.

V. CONCLUSIONS

  • For 1D systems, the energy difference between classically optimized parameters and the true ground state depends only weakly on system size, while 2D relative energy errors increase with lattice size.The paper frames VTNE as potentially performing most or all optimization classically in 1D, but as a VQE pre-optimization method in 2D.Source: §V · p. 7
  • For the 4×2 Hubbard model, Figure 7 plots mean relative energy error against optimization step for random, non-interacting-to-interacting, and MPS-assisted VQE initializations.The plotted means are averaged over 10 sampled parameter sets; panel (b) uses MPS-assisted optimizations at specified bond dimensions.Source: Figure 7 · p. 8
  • The conclusions describe an approximate tensor-network backend as supporting accurate ground-state energy estimation and efficient circuit initialization for large system sizes.The authors state that the approach stands to enhance VQE's scalability and feasibility on near-term hardware.Source: §V · p. 8
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