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Mutual information-assisted Adaptive Variational Quantum Eigensolver

Zi-Jian Zhang, Thi Ha Kyaw, Jakob S. Kottmann, Matthias Degroote, Alán Aspuru-Guzik

arXiv:2008.07553v3quant-ph

TL;DR

Adaptive VQE faces a costly entangler-selection problem, especially when pools grow with system size. The paper screens entanglers using mutual information estimated from a classically approximated ground state. In an H2 setting, the reported pool can be reduced to 1.5% of its original size, and the authors call for more in-depth benchmarks and studies.

  • Problem

    Adaptive methods can require many circuit runs, while QCC entangler pools grow exponentially with the number of qubits.

    Source: §I · p. 1, A. Adaptive ansatz construction in VQE · p. 2

  • Method

    The method estimates pairwise qubit mutual information from a classical ground-state approximation, averages it over each entangler's qubits, and screens the pool by score.

    Source: B. Mutual information-assisted screening · pp. 2–3

  • Results

    1.5% of the original entangler pool suffices in some H2 settings with BK αβαβ, where high correlation strength strongly correlates with high energy descent.

    Source: A. H2 and LiH molecules · p. 4

  • Takeaways & Limitations

    The authors report that MI-assisted adaptive VQE can use much less circuit work, with entanglers mainly placed among qubits with high MI.

    Source: §IV · p. 7

    More in-depth benchmarks and studies are required, and screening rates vary with the fermion-qubit transformation.

    Source: A. H2 and LiH molecules · p. 5, §IV · p. 7

Abstract

from arXiv · show

Adaptive construction of ansatz circuits offers a promising route towards applicable variational quantum eigensolvers on near-term quantum hardware. Those algorithms aim to build up optimal circuits for a certain problem and ansatz circuits are adaptively constructed by selecting and adding entanglers from a predefined pool. In this work, we propose a way to construct entangler pools with reduced size by leveraging classical algorithms. Our method uses mutual information between the qubits in classically approximated ground state to rank and screen the entanglers. The density matrix renormalization group method is employed for classical precomputation in this work. We corroborate our method numerically on small molecules. Our numerical experiments show that a reduced entangler pool with a small portion of the original entangler pool can achieve same numerical accuracy. We believe that our method paves a new way for adaptive construction of ansatz circuits for variational quantum algorithms.

I. INTRODUCTION

VQE is considered promising for near-term quantum chemistry, but NISQ hardware limits and costly ansatz construction constrain its practicality. The paper proposes using classical mutual-information estimates to reduce adaptive entangler pools and circuit runs.

I. INTRODUCTION

  • NISQ limits on qubit counts and circuit depth make it pressing to identify suitable problems and algorithms; VQE is viewed as promising for electronic-structure problems.The passage describes NISQ devices as limited to a few tens of qubits and gates.Source: §I · p. 1
  • Hardware-efficient and UCC ansatzes can require too many gates, making circuits too deep for noisy devices with short coherence times.Source: §I · p. 1
  • Adaptive ansatz methods can reduce gate counts by placing gates selectively, but many circuit runs can negate their computational efficiency.Source: §I · p. 1
  • Mutual information between spin-orbitals has been used to improve classical chemistry algorithms, including DMRG orbital mappings.Even a half-converged DMRG calculation can provide useful MI estimates that help reduce resources needed for DMRG convergence.Source: §I · p. 1
  • The proposed method uses approximate classical MI to reduce entangler pools, and molecular simulations report fewer circuit runs and speed-up.The experiments cover H2, LiH, H2O, and BeH2 Hamiltonians.Source: §I · p. 1

A. Adaptive ansatz construction in VQE

Adaptive ansatz construction builds a VQE trial circuit stepwise by selecting entanglers from a pool. The section motivates this approach by its potential to avoid irrelevant gates, while noting that large pools make selection costly.

A. Adaptive ansatz construction in VQE

  • At each construction step, entanglers from a predefined pool are scored and one is selected; this work scores them by the energy reduction they can provide.Source: A. Adaptive ansatz construction in VQE · p. 2
  • QCC pools can grow exponentially with the number of qubits because they include exponentiated Pauli words, making pool reduction important for preserving accuracy while reducing selection work.Source: A. Adaptive ansatz construction in VQE · p. 2
  • Adaptive methods iteratively add gates in useful positions, avoiding irrelevant operations and potentially reducing gate counts compared with fixed ansatz methods.Source: A. Adaptive ansatz construction in VQE · p. 2
  • The proposed workflow reduces the entangler pool before using it for adaptive ansatz construction.Figure 1 depicts the procedure as classical MI calculation followed by adaptive construction with an MI-reduced pool.Source: Figure 1 · p. 2

B. Mutual information-assisted screening

The screening method uses pairwise qubit mutual information to score entanglers and retain a high-scoring fraction of the original pool. It estimates MI from a classically approximated ground state, using DMRG in the stated protocol.

B. Mutual information-assisted screening

  • Mutual information between two qubits is defined using their individual reduced density matrices and their joint reduced density matrix, with the rest of the qubits traced out.Source: B. Mutual information-assisted screening · p. 2
  • Correlation strength is used as the entangler score, with larger values ranking ahead of smaller ones for screening.The score averages the MI of qubit pairs within the entangler's acted-on subset.Source: B. Mutual information-assisted screening · pp. 2–3
  • The protocol estimates ground-state pairwise MI with DMRG, scores entanglers by average MI among the qubits they act on, and keeps the top pcut percent.It then discards unselected entanglers and performs adaptive construction with the screened pool.Source: B. Mutual information-assisted screening · p. 3

III. NUMERICAL EXAMPLES

The section evaluates MI-assisted adaptive VQE across molecular systems and defines percentile-based screening rates for estimating how much of a complete entangler pool can be screened. It relates the screening rate to circuit-run savings and describes the simulation settings.

III. NUMERICAL EXAMPLES

  • Experiments cover H2, LiH, H2O, and BeH2 with QCC pools, plus a qubit fermionic-excitation pool for BeH2, using JW, parity, and BK mappings.Even-Y entanglers are screened because they commute with the Hamiltonian in the systems without an external magnetic field; Table I details the simulated systems.Source: §III · p. 3, Table I · p. 4
  • Chemical accuracy is defined as 1 milihartree from the FCI energy within the chosen basis set and active space; simulations use DMRG and basin-hopping parameter optimization.The reported optimization settings are 10 iterations, 0.5 temperature, and 10−6 step size.Source: §III · p. 3
  • pmax and pavg summarize selected-entangler percentile ranks; setting pcut above pmax preserves the entangler selections made with the complete pool.pmax is the highest selected percentile, while pavg averages the percentiles of selected entanglers.Source: §III · p. 3
  • pmax implies pmaxN(Eoriginal) trials per adaptive step and Nmax = pmaxN(Eorigin)Nent trials overall, while matching original selection with an average-sized pool may require adjustment at each step.The section expects an adjusted pool between Navg and Nmax could recover original selection using more sophisticated techniques.Source: §III · pp. 3–4
  • Figure 2 plots pmax against bond length for H2 and LiH; using a screened pool of pmaxN(Eoriginal) entanglers reduces circuit runs to approximately pmax percent of the complete-pool count.The qubit count for each setting is given in parentheses.Source: Figure 2 · p. 4
  • For H2, LiH, and H2O, complete-pool runs examine selected-entangler percentiles; acceptable entanglers exceed 30% of the largest energy descent, while BeH2 uses a different selection strategy.Calculations terminate upon reaching chemical accuracy.Source: §III · p. 4

A. H2 and LiH molecules

H2 and LiH screening rates vary with molecular geometry and encoding, with some settings permitting substantial reductions in the entangler pool. The authors recommend choosing transformations according to the system, while noting that rates for many high-screening cases use pools already reduced by stationary-qubit removal.

A. H2 and LiH molecules

  • The runs use pools of 120, 496, 2,016, 8,128, or 32,640 entanglers for 4, 5, 6, 7, or 8 qubits, respectively.The pool size depends on the fermion-to-qubit transformation and qubit count.Source: A. H2 and LiH molecules · p. 4
  • For H2 with BK αβαβ, pmax is around 1.5% at some bond lengths, and pavg is less than half of pmax in most cases.Figure 3 plots pavg against bond length for H2 and LiH; the caption says low pavg implies selected entanglers tend to have high correlation strength overall.Source: A. H2 and LiH molecules · p. 4, Figure 3 · p. 5
  • Screening rates differ across transformations and bond lengths, and pmax can exceed 50% in some cases.For high-screening cases, rates are often calculated relative to pools already reduced by removing stationary qubits.Source: A. H2 and LiH molecules · pp. 4–5
  • Against the original pools, BK αβαβ has the lowest H2 pmax at bond lengths ≤1.3 Å, while parity ααββ is lowest for LiH and longer-bond H2.The authors note that H2 orbital reordering between 1.3 Å and 1.4 Å might partly explain changes in pmax, and recommend system-dependent transformations and further benchmarks.Source: A. H2 and LiH molecules · p. 5

B. H2O molecule

For H2O, both screening rates decrease as bond length grows, reaching low values in the stretched system. The percentile estimates are comparatively stable for early selected entanglers even when the DMRG calculation is unconverged, though deviations grow at high percentiles.

B. H2O molecule

  • At a bond length of 2.4 Å, H2O has pmax 8.44% and pavg 2.44%, indicating substantial pool-size reduction in the stretched system.The study uses the parity transformation and adds a spin-penalty term for bond lengths ≥1.8 Å.Source: B. H2O molecule · p. 6
  • For H2O, pmax and pavg both decrease as bond length increases.Figure 4(a) presents the screening rates across bond lengths.Source: B. H2O molecule · p. 6
  • At 1.8 Å, percentile differences among converged and unconverged DMRG estimates are not significant for the first 16 entanglers.The converged calculation is within 1 milihartree of FCI; the two unconverged calculations are about 10 and 20 milihartree away.Source: B. H2O molecule · p. 6
  • In the worst case at 1.8 Å, unconverged DMRG raises pmax from 17.5% to 27.2%, which the authors regard as a good MI estimate despite its energy deviation.The deviation becomes apparent at high percentiles.Source: B. H2O molecule · p. 6

C. BeH2 molecule

On BeH2, screening has different circuit-depth effects for the qubit fermionic-excitation and QCC pools. The authors report similar chemical-accuracy depths across tested cutoffs for the former, while larger QCC pools yielded fewer entanglers to convergence.

C. BeH2 molecule

  • The tested BeH2 pools contain 4,137 qubit fermionic-excitation entanglers and 2.10 × 10^6 QCC entanglers before screening.The tested cutoffs select 1,002, 1,638, or 4,137 qubit fermionic-excitation entanglers and 4,102, 12,544, or 20,906 QCC entanglers.Source: C. BeH2 molecule · p. 6
  • For the qubit fermionic-excitation pool, chemical-accuracy circuits use 31, 29, and 30 entanglers at pcut = 20%, 40%, and 100%, respectively.The authors say excluding some entanglers with larger energy descent does not significantly affect the required circuit depth.Source: C. BeH2 molecule · p. 7
  • For the QCC pool, converged circuits use 37, 32, and 28 entanglers at pcut = 0.2%, 0.6%, and 1.0%, respectively.The authors report that final circuit depths decrease with the larger QCC pools.Source: C. BeH2 molecule · p. 7
  • The converged circuit uses 31 UCC entanglers, equivalent to 200 QCC entanglers, compared with 28–37 entanglers in the screened QCC runs.The UCC pool contains 345 entanglers; the authors report shallower circuits than in the previous UCC-pool work.Source: C. BeH2 molecule · p. 7

IV. CONCLUSIONS & DISCUSSION

The conclusion presents mutual-information-based entangler-pool reduction as a way to support adaptive ansatz construction with fewer circuit runs. It reports low average screening rates while noting that cutoff choice and the relationship to classical simulation difficulty remain unresolved.

IV. CONCLUSIONS & DISCUSSION

  • The proposed mutual-information method reduces entangler-pool size and allows adaptive ansatz construction with much less circuit runs.Mutual information can be approximated by classical algorithms such as DMRG.Source: §IV · p. 7
  • Low pavg indicates entanglers should be placed mainly among qubits with high mutual information, and screening rates vary across fermion-qubit transformations.The authors describe screening rates as an additional criterion for quantifying transformation performance.Source: §IV · p. 7
  • There is no systematic way to predict pmax or set pcut; suggested heuristics include accuracy, supported circuit depth, runtime, stability, and selected-entangler percentiles.A larger pcut may be considered when many selected entanglers have percentiles near the cutoff.Source: §IV · p. 7
  • Because pmax can be nearly continuous across molecular geometries, suitable pcut values might transfer to systems expected to behave similarly.This is presented as a possibility when suitable pcut values are known for a range of chemical systems.Source: §IV · p. 7
  • Whether low screening percentiles also correspond to lower classical simulation difficulty remains unknown.The authors identify connecting pmax and pavg to classical simulation resource counts as an open question.Source: §IV · p. 7
  • Classical mutual-information precomputation may guide qubit mapping onto noisy hardware according to connectivity and entanglement capabilities.The proposed mapping is to physical qubits in an optimal hardware configuration [43].Source: §IV · p. 7

Appendix A

Appendix A includes figures of pmax and pavg screening rates for LiH and H2, each stated relative to its original entangler pool.

Appendix A

  • Figure 6 gives LiH screening rates pmax and pavg relative to the 6-qubit original pool of 2016 entanglers.The caption identifies the system and pool scope but supplies no plotted values in the provided passage.Source: Figure 6 · p. 10
  • Figure 7 gives H2 screening rates pmax and pavg relative to the 8-qubit original pool of 32640 entanglers.The caption identifies the system and pool scope but supplies no plotted values in the provided passage.Source: Figure 7 · p. 10
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