PaperlayerView exact revision on arXiv

Source-linked AI summary

Variational Quantum Eigensolver with Fewer Qubits

Jin-Guo Liu, Yi-Hong Zhang, Yuan Wan, Lei Wang

arXiv:1902.02663v2quant-phcond-mat.str-el

TL;DR

Exact classical contraction of finite PEPS has exponential complexity, motivating quantum estimation for two-dimensional area-law states. The paper uses qubit-efficient, symmetry-aware tensor-network circuits and reports a 0.97 fidelity for a 4 × 4 frustrated Heisenberg model, while noting measurement-and-reset timing constraints.

  • Problem

    Exact classical contraction of finite PEPS has exponential complexity with system size, despite PEPS representing many two-dimensional area-law states with polynomially many parameters.

    Source: §I · p. 1

  • Method

    The scheme prepares MPS or PEPS with fewer qubits through sequential measurement and reuse, using gradient-based, quantum-number-preserving circuit designs.

    Source: §I · pp. 1–2, §II · p. 3

  • Results

    0.97 fidelity was reached with variational energy −0.463 using the SU(2)-symmetric circuit with M = 300 parameters.

    Source: A. MPS inspired ansatz with conserved… · p. 5

  • Takeaways & Limitations

    The proposed scheme accesses ground-state properties with fewer qubits, and symmetry-based circuit design can alleviate vanishing gradients and speed convergence.

    Source: §V · p. 8

    On SQUID devices, measurement and reset can take several microseconds, slower than applying a gate.

    Source: §IV · p. 8

Abstract

from arXiv · show

We propose a qubit efficient scheme to study ground state properties of quantum many-body systems on near-term noisy intermediate scale quantum computers. One can obtain a tensor network representation of the ground state using a number of qubits smaller than the physical degrees of freedom. By increasing the qubits number, one can exponentially increase the bond dimension of the tensor network variational ansatz on a quantum computer. Moreover, we construct circuits blocks which respect U(1) and SU(2) symmetries of the physical system and show that they can significantly speed up the training process and alleviate the gradient vanishing problem. To demonstrate the feasibility of the qubit efficient variational quantum eigensolver in a practical setting, we perform first principle classical simulation of differentiable programming of the circuits. Using only $6$ qubits one can obtain the ground state of a $4\times 4$ square lattice frustrated Heisenberg model with fidelity over 97%. Arbitrarily long ranged correlations can also be measured on the same circuit after variational optimization.

I. INTRODUCTION

The introduction frames ground-state VQE as promising but constrained by qubit resources and vanishing gradients. It proposes qubit-efficient MPS/PEPS circuits and symmetry-aware, gradient-based training for programmable gate-model devices.

I. INTRODUCTION

  • Unstructured randomly parameterized circuits can have gradients that vanish exponentially with parameter count, while standard VQE requires as many qubits as the problem size.The introduction links gradient concentration to exponentially small overlap between a random initial state and the target state.Source: §I · p. 1
  • Sequential measurement and reuse generate MPS on a quantum computer using fewer qubits than physical degrees of freedom, with access to bond dimensions inaccessible to classical computers.The introduction notes that small-bond-dimension MPS can be simulated classically and that MPS can represent systems with diverse lattice geometries and topologies.Source: §I · p. 1
  • The proposed PEPS-inspired circuit targets two-dimensional area-law states, whose exact classical contraction has exponential complexity, while quantum hardware can estimate its energy efficiently.The PEPS representation uses polynomially many parameters.Source: §I · p. 1
  • On programmable gate-model computers, systematic control of bond dimension and variational parameters combines with gradient-based learning and quantum-number-preserving circuits to speed training.The authors distinguish this setup from prior cavity-QED continuous-MPS optimization with few tuning parameters.Source: §I · pp. 1–2
  • The paper applies the approach to a frustrated Heisenberg model, estimates experimental gate-count timing, and discusses future research directions.The paper also provides codes and pretrained circuit parameters in a repository.Source: §I · p. 2

II. CIRCUIT ARCHITECTURE AND TRAINING APPROACH

The method prepares and samples an MPS or PEPS for an N-qubit system with substantially fewer qubits by sequentially measuring and reusing physical qubits. It estimates energy gradients with parameter shifts and trains the circuit using a classical optimizer.

II. CIRCUIT ARCHITECTURE AND TRAINING APPROACH

  • An N-qubit variational state is represented by a parametrized circuit, and its energy is the sum of Hamiltonian-term expectation values.Commuting Hamiltonian terms can be measured together in corresponding bases.Source: §II · p. 2
  • Using R + V ≪ N qubits, the scheme treats R qubits as physical and V as virtual degrees of freedom, then sequentially measures and reuses physical qubits to sample an MPS or PEPS.For Q-MPS, R = 1; after the repeated measure-reset steps, the remaining qubits are measured to collect the final outcomes.Source: §II · p. 2
  • The qubit-efficient scheme supports arbitrarily long-range interactions and fermionic systems through the Jordan-Wigner transformation, although quantum chemistry Hamiltonians contain O(N^4) terms.Measurement counts can be reduced using cited techniques.Source: §II · p. 2
  • For circuit parameters of the form e^-iθ_iΣ/2 with Σ^2 = 1, shifting θ_i by ±π/2 gives an exact gradient estimate for noisy stochastic optimization.The gradient information is passed to a classical optimizer, which updates parameters until a convergence criterion is met.Source: §II · p. 3

III. APPLICATION TO HEISENBERG MODEL

The application section motivates the study of frustrated spin models and specifies a 4 × 4 open square-lattice Heisenberg example. It evaluates energy and gradients by sampling three Pauli bases.

III. APPLICATION TO HEISENBERG MODEL

  • The study considers an open 4 × 4 square lattice with frustration strength J2 = 0.5, where J2 suppresses Néel order.The sites are zigzag-ordered in the ansatz.Source: §III · p. 3
  • Energy and gradient estimates for the model are obtained by sampling circuit outputs in the σx, σy, and σz bases.The Hamiltonian includes nearest- and next-nearest-neighbor pairs.Source: §III · p. 3
  • Classical approaches to frustrated quantum spin models face the sign problem or high computational cost at larger bond dimensions.The authors present Q-MPS/Q-PEPS optimization on near-term quantum computers as a promising approach to these open problems.Source: §III · p. 3

A. MPS inspired ansatz with conserved quantum numbers

The ansatz reuses measured physical qubits while virtual qubits mediate entanglement, with U(1)- and SU(2)-preserving blocks tailored to the Heisenberg model. In the reported 4×4 study, the SU(2) circuit achieves the highest fidelity with fewer parameters, while gradient behavior depends on system size and virtual-qubit count.

A. MPS inspired ansatz with conserved quantum numbers

  • U(1) blocks preserve S_z by using symmetry-preserving gates and a spin-balanced initial state; SU(2) blocks prepare singlets with an ancilla and apply parametrized SWAP operations.The singlet preparation uses the ancilla to carry entanglement between odd and even steps.Source: A. MPS inspired ansatz with conserved… · pp. 3–4
  • The circuit uses one physical qubit that is measured and reused, while V virtual qubits mediate entanglement; the SU(2) construction adds an ancilla for singlet initialization.For the 4×4 model, V = 4 and the circuit has 12 sequential physical-qubit uses.Source: Figure 2 · p. 4
  • 0.97 fidelity (SU(2), 300 parameters) exceeds 0.92 (U(1)) and 0.69 (unstructured), while SU(2) variational energy reaches −0.463.The SU(2) ansatz gives the best variational energy despite using fewer parameters than the other reported blocks.Source: A. MPS inspired ansatz with conserved… · p. 5
  • The SU(2) circuit produces fidelity 0.98 for the unfrustrated lattice, where the checkerboard antiferromagnetic correlation pattern is more visible than in the frustrated case.The same SU(2) circuit is used to measure spin-spin correlations in both cases.Source: A. MPS inspired ansatz with conserved… · p. 5, Figure 5 · p. 5
  • For an open Heisenberg chain with V = 4, gradient variance decays by a power law with system size, rather than the exponential decay reported for generic circuits.The authors attribute this behavior to the low-entropy ansatz capturing the target ground state's inductive bias.Source: A. MPS inspired ansatz with conserved… · p. 5
  • For an N = 20 chain, gradient variance decreases exponentially as V grows in the V ≪ N regime, although symmetry-preserving circuits have larger gradients than generic circuits at small V.The gradients remain much smaller than their small-V values, indicating that increasing V makes the Q-MPS harder to train.Source: A. MPS inspired ansatz with conserved… · pp. 5–6
  • The authors caution that larger bond dimensions require exponentially more parameters to offset vanishing gradients, while sampling error can obscure very small gradient signals.They identify parameter-efficient design as necessary for scaling to highly entangled two-dimensional systems.Source: A. MPS inspired ansatz with conserved… · p. 6

B. PEPS inspired ansatz

The section presents a PEPS-inspired quantum ansatz for two-dimensional systems, designed to capture area-law entanglement without requiring exact classical PEPS contraction. Simulations show similar or slightly better performance than Q-MPS at 4×4 and a lower ground-state energy at 6×6, while gradient and sampling behavior remains a scaling concern.

B. PEPS inspired ansatz

  • PEPS can represent many area-law-entangled high-dimensional states with polynomially many parameters, but exact contraction for energy expectation values has exponential complexity.The authors describe their circuit as PEPS-inspired because an efficient scheme to prepare Q-PEPS on a quantum circuit is unavailable in their discussion.Source: B. PEPS inspired ansatz · p. 6
  • The ansatz assigns a constant number of virtual qubits per physical qubit and entangles physical and virtual qubits in repeated layers, yielding two-dimensional area-law entanglement.Physical qubits can be measured in parallel.Source: B. PEPS inspired ansatz · p. 6
  • At 4×4, Q-PEPS performs similarly or slightly better than Q-MPS; at 6×6, it obtains a much lower ground-state energy.The 4×4 Q-PEPS uses V/R = 1 and depth 5, with 180 parameters, matching a depth-3 Q-MPS.Source: B. PEPS inspired ansatz · p. 6
  • At batch size 4096, rgs is 0.142/0.021 at 4×4 and 0.088/0.032 at 6×6, with gradient decreasing and sampling error increasing.When rgs is much less than 1, gradient information can be smeared out and training may fail; the authors say a quantum device is needed to verify scalability.Source: B. PEPS inspired ansatz · p. 6

IV. COMPLEXITY ANALYSIS BY GATE COUNTING

The gate-count analysis contrasts exponential classical simulation cost with constant device gate-time scaling in qubit number, while identifying depth and measurement/reset overheads as practical constraints. A 4 × 4 frustrated Heisenberg model reaches fidelity 0.917 with 60 parametrized gates.

IV. COMPLEXITY ANALYSIS BY GATE COUNTING

  • Actual-device gate time scales constantly with qubit number, while classical-simulation gate time scales exponentially with virtual qubits V.For d = 1, the 4 × 4 Heisenberg model is estimated to take approximately 18 minutes on one QPU, excluding readout time.Source: §IV · p. 7
  • Fidelity 0.917 is reached with 60 parametrized gates for the 4 × 4 frustrated Heisenberg model at J2 = 0.5.The authors describe this depth as within reach of current-day quantum technology.Source: Table III · p. 7, §IV · p. 7
  • Parallel QPUs could distribute gradient estimation, with classical communication used to collect gradients measured across devices.The paper presents this as an anticipated route to high-throughput VQE, not a demonstrated result.Source: §IV · p. 7
  • For larger systems, circuit depth must grow linearly with N, and under area-law entanglement the circuit width must grow linearly with boundary size.The total running time MTgate must remain below qubit coherence time, constraining near-term ansätze to shallow circuits.Source: §IV · p. 7
  • Measurement and reset latency is device dependent; on SQUIDs, a single measure-and-reset can take several microseconds, slower than a gate.Using the architecture without qubit reuse still retains the enhanced gradient signal.Source: §IV · p. 8

V. DISCUSSIONS

The discussion presents qubit-efficient VQE as a way to study ground states with fewer qubits than physical degrees of freedom. It highlights symmetry-based circuit design and points to excited-state sectors and translational symmetry as further directions.

V. DISCUSSIONS

  • The proposed qubit-efficient VQE accesses ground-state properties using fewer qubits than the system size.The discussion frames classical quantum many-body computation as a source of insight for quantum algorithms.Source: §V · p. 8
  • Exploiting physical symmetries in circuit design can alleviate gradient vanishing and speed convergence to the ground state.Source: §V · p. 8
  • Conserved quantum numbers can provide access to excited states in different sectors, while parameter sharing may incorporate spatial translational symmetry.The discussion raises symmetry-constrained universal gate sets as an open question.Source: §V · p. 8

Appendix A: Qubit efficient scheme for cluster state

The appendix demonstrates preparation of a one-dimensional cluster state with qubit reuse and relates it to an MPS of bond dimension two. It also describes how local filtering and conditional measurements yield a nonzero second-nearest-neighbor correlation for estimation.

Appendix A: Qubit efficient scheme for cluster state

  • The N-qubit 1D cluster state is prepared by initializing qubits in |+⟩ and applying CZ gates sequentially between nearest neighbors.Source: Appendix A · p. 10
  • The cluster state has an equivalent MPS representation with bond dimension two and can be prepared using two qubits by measuring, resetting, and reusing the physical qubit.A SWAP exchanges its state with the virtual qubit between preparation steps.Source: Appendix A · p. 10
  • Two-point Pauli correlations vanish on the unfiltered 1D cluster state, whereas applying the stated SLOCC operation yields a second-nearest-neighbor correlation of cos 2θ sin γ.Source: Appendix A · p. 10
  • To estimate ⟨σz_2σz_4⟩, measure qubits 2 and 4 in Z, condition on outcome 0 after applying HRz(π/2) to qubit 3, then randomly choose remaining measurement bases.Repeat the protocol until the estimate is adequate.Source: Appendix A · p. 10
Loading 1902.02663v2…