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Adaptive pruning-based optimization of parameterized quantum circuits

Sukin Sim, Jonathan Romero, Jerome F. Gonthier, Alexander A. Kunitsa

arXiv:2010.00629v1quant-ph

TL;DR

Optimizing large parameter spaces remains a challenge for parameterized quantum circuits, even as expressive ansatze have been developed. PECT iteratively optimizes and reallocates subsets of ansatz parameters; in molecular VQE examples, it reduced circuit resources and estimated runtime, including a 73% reduction in estimated LDCA runtime for LiH.

  • Problem

    Optimizing parameterized quantum circuits remains challenging because larger circuits burden optimizers with vast parameter spaces and many function evaluations.

    Source: §I · p. 1, §II · p. 2

  • Method

    PECT repeatedly optimizes a subset of a predefined ansatz’s parameters and reallocates the subset between iterations using pruning and regrowth.

    Source: §I · p. 1, §II · p. 4, A. Parameter reallocation for… · p. 4

  • Results

    For LiH LDCA VQE, PECT reduced estimated optimization runtime by 73% on average across bond lengths, through reductions in circuit depth and function calls.

    Source: §I · p. 2

  • Takeaways & Limitations

    In noiseless simulations of small molecular systems, PECT may help converge difficult high-parameter redundant ansatze while reducing circuit depth, two-qubit gate counts, and optimization runtimes.

    Source: §VI · p. 17

    The reported demonstrations are noiseless VQE simulations on small molecular systems.

    Source: §VI · p. 17

Abstract

from arXiv · show

Variational hybrid quantum-classical algorithms are powerful tools to maximize the use of Noisy Intermediate Scale Quantum devices. While past studies have developed powerful and expressive ansatze, their near-term applications have been limited by the difficulty of optimizing in the vast parameter space. In this work, we propose a heuristic optimization strategy for such ansatze used in variational quantum algorithms, which we call "Parameter-Efficient Circuit Training" (PECT). Instead of optimizing all of the ansatz parameters at once, PECT launches a sequence of variational algorithms, in which each iteration of the algorithm activates and optimizes a subset of the total parameter set. To update the parameter subset between iterations, we adapt the dynamic sparse reparameterization scheme by Mostafa et al. (arXiv:1902.05967). We demonstrate PECT for the Variational Quantum Eigensolver, in which we benchmark unitary coupled-cluster ansatze including UCCSD and k-UpCCGSD, as well as the low-depth circuit ansatz (LDCA), to estimate ground state energies of molecular systems. We additionally use a layerwise variant of PECT to optimize a hardware-efficient circuit for the Sycamore processor to estimate the ground state energy densities of the one-dimensional Fermi-Hubbard model. From our numerical data, we find that PECT can enable optimizations of certain ansatze that were previously difficult to converge and more generally can improve the performance of variational algorithms by reducing the optimization runtime and/or the depth of circuits that encode the solution candidate(s).

I. INTRODUCTION

Variational quantum algorithms depend on optimizing parameterized circuits, but large parameter spaces make optimization difficult while practical ansatze should be expressive, low-depth, and tractable to optimize. PECT addresses this by repeatedly optimizing subsets of an ansatz’s parameters and reallocating the subset between iterations.

I. INTRODUCTION

  • PECT targets redundant parameterizations by optimizing a subset of an ansatz’s parameters and refining both the active gates and parameter values without prior knowledge of the target state.The paper considers UCCSD, k-UpCCGSD, and LDCA as examples of ansatze with potentially large parameter counts.Source: §I · p. 1
  • Scaling variational algorithms to larger circuits burdens classical optimizers with many function evaluations, while practical circuits should also be low-depth, accurate, and relatively easy to optimize.VQAs use measured expectation values to evaluate an objective that a classical optimizer varies through circuit parameters.Source: §I · p. 1, §II · p. 2

II. PARAMETER-EFFICIENT CIRCUIT TRAINING (PECT)

  • PECT assumes the ansatz can express the solution while containing removable parameterized gates, and that setting a gate’s parameter to zero implements the identity.The paper notes that redundant single-qubit rotations can occur in commonly used circuit patterns.Source: §II · p. 2
  • Global sparsity controls the active subset size, and parameters outside that subset are replaced by identity operations, producing shorter circuits in depth and operation count.The initial subset is sampled across circuit layers; local parameter values may be randomly initialized or informed by guesses such as MP2 amplitudes.Source: §II · pp. 3–4
  • Each iteration locally optimizes the active subset, then reallocates parameters until termination criteria are met, yielding an optimized ansatz substructure as the solution candidate.Local optimization can use a classical optimizer such as L-BFGS-B; termination criteria include objective convergence, a function-call limit, or oscillation in recent objective values.Source: §II · p. 4

A. Parameter reallocation for parameterized quantum circuits

PECT reallocates the active circuit parameters by pruning small-magnitude values, adapting the pruning threshold toward a target pruning count, and regrowing parameter slots across layers.

A. Parameter reallocation for parameterized quantum circuits

  • The user sets Np, the initial threshold H(0), and δ; Np controls how much the subset is shuffled, and δ was fixed at 0.1 in the reported simulations.The initial threshold was chosen to avoid excessive first-iteration pruning; preliminary calculations found no significant performance effect from δ.Source: A. Parameter reallocation for… · p. 4
  • At each reallocation, PECT removes active parameters whose magnitudes fall below the current threshold, then adjusts the next threshold according to whether the total pruned count misses the target Np.The implementation lowers the threshold when too many are pruned and raises it when too few are pruned.Source: A. Parameter reallocation for… · pp. 4–5
  • PECT regrows as many parameter slots as were pruned, allocating more to layers with more surviving parameters while keeping global sparsity constant.If a layer’s updated allocation exceeds its parameter count, excess slots are redistributed randomly to other layers.Source: A. Parameter reallocation for… · p. 5
  • Survivor-based growth treats layers with fewer pruned parameters as more important, but whether they are genuinely more important remains to be investigated.The authors suggest larger gradient magnitudes as one possible indicator of layer importance.Source: A. Parameter reallocation for… · p. 5

B. Prior work

PECT combines adaptive ansatz construction with pruning, but operates on an ordered, predefined ansatz structure and can prune and regrow multiple gates per iteration.

B. Prior work

  • Unlike ADAPT-VQE, which grows an ansatz from an unordered operator pool one operation at a time using gradients, PECT refines a subset of gates from a predefined ordered structure.PECT can prune and grow multiple gates in a single iteration.Source: B. Prior work · p. 5
  • Unlike Ansatz Architecture Search, which ranks architectures formed by removing one two-qubit operation, PECT prunes gates according to optimized parameter values and a threshold.PECT also includes an ansatz-growth phase to reallocate parameters and gates.Source: B. Prior work · p. 5
  • PECT brings together adaptive construction and pruning to seek a compact solution-state representation within a predefined ansatz structure.The paper contrasts this objective with ADAPT-VQE’s growth from an unordered pool and AAS’s architecture-ranking procedure.Source: B. Prior work · p. 5

III. NUMERICAL EXPERIMENTS

The numerical study uses VQE to optimize three ansatze, with the premise that optimizing parameter subsets can reduce the work of each run. Activating fewer parameters is also expected to produce circuits with fewer gates and lower depth.

III. NUMERICAL EXPERIMENTS

  • Although a sequence of VQE runs may seem costly, each run may have a shorter optimization runtime because fewer circuit parameters need exploration and updating.The stated runtime advantage is a rationale for the approach, not a numerical result in these passages.Source: §III · p. 5
  • The numerical experiments use VQE to optimize UCCSD, k-UpCCGSD, and LDCA circuits.The paper presents VQE as its example variational quantum algorithm for demonstrating PECT.Source: §III · pp. 5–6
  • Activating only a fraction of the parameters likely yields circuits with fewer gate operations and lower depth.The paper also notes that a measurement budget could be imposed for each local optimization, though that idea was not used in its simulations.Source: §III · p. 6

A. Ansatze

The ansatze differ in circuit depth, parameter count, and optimization demands, shaping their potential suitability for NISQ-era applications. UCCSD is deep but compact for the small systems considered, whereas k-UpCCGSD and LDCA offer shallower circuits with different parameterization trade-offs.

A. Ansatze

  • UCCSD circuits have depth scaling as O(N^5), while k-UpCCGSD has depth O(kN) and was developed as a shallower, more flexible UCC variant.The source also notes that low-rank decomposition and fermionic swap networks can reduce UCCSD depth by approximately a linear factor under linear connectivity [48].Source: A. Ansatze · p. 6
  • LDCA uses repeated nearest-neighbor parameterized two-qubit gates and has linear depth scaling, but its compact nested structure accrues many parameters.Its gates are native to devices with tunable couplers.Source: A. Ansatze · p. 6
  • For the studied 12-qubit instances, UCCSD used 56 parameters, 2-UpCCGSD 60, and five-superlayer LDCA 1662 in its original formulation.The study's LDCA implementation uses parameter sharing, reducing each two-qubit block from five free parameters to three.Source: A. Ansatze · p. 6
  • For these small systems, UCCSD can have fewer parameters despite its larger depth; it also offers informed initialization techniques and relative ease of optimization.The study defines UCCSD as one PECT layer, k-UpCCGSD as k layers, and each LDCA two-qubit block as a layer.Source: A. Ansatze · p. 6

B. Computational details

The study compares PECT and full-ansatz VQE optimizations on LiH and water using matched optimizers and initial parameters. On LiH, PECT made LDCA optimization feasible and an alternative optimizer brought its errors below chemical accuracy across the tested bond lengths.

B. Computational details

  • The VQE comparison covered LiH and symmetric O–H stretching in H2O, using non-PECT full-ansatz optimization and PECT with matched classical optimizers and initial parameters.The listed Table I reports hyperparameters and parameter counts for the PECT calculations.Source: B. Computational details · pp. 6–7, Table I · p. 7
  • PECT hyperparameters were selected using ansatz parameter counts and initial guesses; the fractional tolerance δ was fixed at 0.1.For LiH UCCSD, H0 was set slightly below the average magnitude of the MP2 amplitudes to limit early pruning.Source: B. Computational details · p. 7

C. LiH

  • For LiH, non-PECT UCCSD reached the highest energy accuracy but used the deepest circuits; PECT reduced its depth by up to approximately 20% in some cases, with some accuracy loss.PECT also reduced 2-UpCCGSD depth, while its energies remained below chemical accuracy; 5–6-superlayer LDCA used 1002–1200 parameters before pruning.Source: C. LiH · p. 7
  • PECT enabled LDCA optimizations that did not converge without it; SLSQP with PECT brought LiH errors below chemical accuracy at all tested bond lengths.This setting used global sparsity 0.6, or around 420 parameters, and about half the function calls of LDCA with L-BFGS-B and PECT.Source: C. LiH · pp. 7–8

D. Symmetric stretching of O-H bonds in H2O

For symmetric O–H stretching in water, PECT reduced circuit depth across the tested ansatze, with accuracy and optimization sensitivity varying by ansatz. LDCA reached chemical accuracy at every tested bond length, though two geometries required six superlayers.

D. Symmetric stretching of O-H bonds in H2O

  • At O–H = 2.4 Å, PECT produced a significantly lower UCCSD energy than full-ansatz optimization, at the same circuit depth.Across the stretched geometries, UCCSD energy estimates worsened; unfavorable Trotterized cluster-operator ordering may also contribute to error.Source: D. Symmetric stretching of O-H bonds… · p. 8
  • PECT reduced k-UpCCGSD circuit depth in every case, but improved accuracy only at some bond lengths; k = 4 and k = 5 were needed for water.Final k-UpCCGSD energies were generally lower than UCCSD in regions where UCCSD errors were high, though circuit depths were comparable or higher.Source: D. Symmetric stretching of O-H bonds… · p. 8
  • For k=5 at stretched O–H distance R=2.4 Å, PECT reached comparable or lower energies using a fraction of circuit depth and comparable function calls, but low-energy convergence remained unreliable.Across ten initial points, one 4-UpCCGSD PECT trial achieved chemical accuracy. First-layer MP2-amplitude initialization performed better than random initialization, but did not ensure reliable convergence.Source: D. Symmetric stretching of O-H bonds… · p. 8
  • PECT-based LDCA reached chemical accuracy for every tested water bond length, using six superlayers at two geometries and five at the others.The five-superlayer circuits had depth 1382 and 3300 two-qubit gates; the two six-superlayer cases had depths around 1658 and about 3960 two-qubit gates.Source: D. Symmetric stretching of O-H bonds… · pp. 8–9
  • LDCA circuits had at least half the depth of the other ansatze but needed substantially more function evaluations, consistent with their larger parameter counts.Gradients require two function evaluations per parameter, and LDCA appeared less sensitive to initialization than k-UpCCGSD under PECT.Source: D. Symmetric stretching of O-H bonds… · p. 9

IV. DISCUSSION

The discussion finds that PECT's robustness and usefulness depend on the ansatz: LDCA is highly prunable, while UCCSD is not well suited to pruning at the studied sizes. It also describes a layerwise extension tested on the one-dimensional Fermi-Hubbard model.

IV. DISCUSSION

  • The discussion proposes combining PECT with layerwise circuit training for larger circuits and tests the combined strategy on one-dimensional Fermi-Hubbard ground-state densities.PECT hyperparameters for water LDCA were based on corresponding LiH values; similar circuit widths and depths may also allow hyperparameters to be inferred across runs.Source: §IV · pp. 9–10

A. Robustness to random parameter initialization in LDCA

  • Across three LiH bond lengths, 20/25, 23/25, and 16/25 random-start LDCA runs reached chemical accuracy while optimizing about 291 of 1002 parameters per local optimization.The success counts correspond respectively to bond lengths 1.5, 2.5, and 3.3 Å; the lower success ratio at 3.3 Å is attributed to increased ground-state description difficulty as the bond stretches.Source: A. Robustness to random parameter… · p. 11

B. Prunability of ansatz

  • PECT appeared most effective for LDCA, whereas k-UpCCGSD remained sensitive to initial parameters and selected gates; in a ten-start 4-UpCCGSD test, only one run reached chemical accuracy.The authors identify LDCA's greater parameter redundancy as a likely explanation for its performance difference from k-UpCCGSD.Source: B. Prunability of ansatz · p. 12
  • For the studied problem sizes, UCCSD was least prunable because its parameter count was already low enough that subset optimization was unnecessary.The authors note that UCCSD's larger challenges include expressibility and Trotterized operator ordering; because PECT assumes fixed operation ordering, ADAPT-VQE may better suit an unordered cluster-operator pool.Source: B. Prunability of ansatz · p. 12

C. Ansatz cost reduction

PECT reduces circuit resources by activating subsets of ansatz parameters, with the size of the reductions depending on the ansatz and molecule. Its runtime trade-offs also vary: it can speed some LiH optimizations but slow others, and water-molecule comparisons are constrained by convergence failures.

C. Ansatz cost reduction

  • Using PECT on LiH reduced circuit depth and two-qubit gate count by 20% each for UCCSD, 28% and 42% for 2-UpCCGSD, and 42% and 58% for LDCA.The respective active parameter counts were 45 of 56, 30 of 60, and 420 of 1002.Source: C. Ansatz cost reduction · p. 12
  • For water, PECT reduced depth and two-qubit gate count by 12% each for UCCSD, 17% and 31% for 4-UpCCGSD, and 17% and 32% for 5-UpCCGSD, averaged across bond lengths.Some geometries did not reach chemical accuracy, limiting equally meaningful resource comparisons.Source: C. Ansatz cost reduction · p. 12
  • For water LDCA, PECT reduced depth by 42% with five superlayers and 39% with six; six were needed at some geometries because five did not reach chemical accuracy.The six-superlayer solutions included gates from the sixth superlayer.Source: C. Ansatz cost reduction · pp. 12–13
  • PECT produced more depth-efficient circuits, but whether that improves noise resilience remains untested and is left for future work.Source: C. Ansatz cost reduction · p. 13

D. Optimization cost estimation

  • The runtime proxy compares optimizations using circuit depth and function-call counts, treating measurement repetitions as constant despite their potentially large real-world cost.PECT's proxy sums depth-weighted calls across local optimizations; it may lower depth per evaluation while requiring more evaluations.Source: D. Optimization cost estimation · p. 13
  • For LiH, PECT reduced estimated runtime for 2-UpCCGSD and LDCA at most or all bond lengths, but full-circuit UCCSD was faster.The plotted runtime proxy is depth multiplied by function calls.Source: D. Optimization cost estimation · p. 13, Figure 6 · p. 14
  • For water, PECT increased UCCSD runtime where both methods reached chemical accuracy; its runtime benefit was more apparent at larger k for k-UpCCGSD.The authors expected LDCA to reduce runtime based on LiH results, but did not run non-PECT LDCA optimizations.Source: D. Optimization cost estimation · pp. 13–14

E. Scaling up: layerwise PECT

Layerwise PECT applies subset optimization sequentially to circuit layers, then compares its Fermi-Hubbard results with prior layerwise training. It reaches comparable energy-density errors with fewer circuit layers, while requiring more function evaluations.

E. Scaling up: layerwise PECT

  • The layerwise procedure optimizes each layer with PECT, freezes its final parameters, then proceeds to the next layer initialized near identity.The near-identity initialization is intended to avoid spending many calls recovering the previous layer's energy; relaxing earlier parameters later can also lead to higher energy.Source: E. Scaling up: layerwise PECT · p. 14
  • Layerwise PECT reached comparable energy densities for different chain lengths with significantly fewer circuit layers, but required more function evaluations before convergence.PECT layers comprised two ansatz sublayers, versus four per layer in the prior layerwise results.Source: E. Scaling up: layerwise PECT · p. 15, Figure 8 · p. 16
  • The trade-off is lower circuit depth versus more function calls; PECT may suit experiments that prioritize depth and can afford those calls.The authors identify depth reduction as especially relevant near term and note measurement-aware strategies as a possible way to mitigate evaluation cost.Source: E. Scaling up: layerwise PECT · p. 15

F. Parameter dynamics of multi-layered PQCs

The authors track average parameter displacement from final values by circuit layer during optimization. Earlier layers move less than later ones, a pattern they connect to the influence of earlier-layer settings on later optimization.

F. Parameter dynamics of multi-layered PQCs

  • Earlier-layer parameters start closer to and stray less from their final values than later-layer parameters, on average.This pattern appeared in both PECT and non-PECT runs, including analyses beyond the illustrated instance.Source: F. Parameter dynamics of multi-layered… · p. 16
  • The displacement analysis used LiH at R = 2.5Å with LDCA and did not employ layerwise training.The authors suggest that earlier-layer settings influence how later-layer parameters are tuned, motivating strategic initialization of early layers.Source: F. Parameter dynamics of multi-layered… · p. 16

V. FUTURE DIRECTIONS

PECT optimizes parameterized circuits by iteratively changing which gates are active, with the aim of improving convergence and reducing circuit resources. The paper also identifies hyperparameter and batching constraints, while pointing to layerwise and other pruning strategies as future directions.

V. FUTURE DIRECTIONS

  • Reducing the active parameter count can lower costs for local optimizers such as quantum natural gradient, whose metric-tensor computation and inversion scale with parameter count.PECT can also be combined with more sophisticated layerwise training schemes.Source: §V · p. 16
  • PECT performance is sensitive to global sparsity and the target number of parameters to prune; excessive sparsity can trigger oscillating-energy termination, while low sparsity can require more function calls.The authors identify better hyperparameter tuning as a direction for making PECT more effective.Source: §V · pp. 16–17
  • For LDCA modeling LiH, earlier layers start and generally remain closer to their final parameter values than later layers.The authors suggest that more sophisticated layerwise training could be used with layerwise PECT.Source: §V · p. 16, Figure 9 · p. 17
  • The current L-BFGS-B and SLSQP implementation does not effectively batch circuits, potentially raising experimental runtime through network latency; batch-supporting optimizers are a proposed alternative.Particle swarm optimization is given as an example of an optimizer supporting batch evaluations.Source: §V · p. 17

VI. CONCLUSION

  • By optimizing only a parameter subset and iteratively reallocating gates, PECT can refine circuits while saving resources and may extend to multiple quantum-computing applications.The authors expect sparse-network pruning methods to help generate more compact quantum circuits.Source: §VI · pp. 17–18, §V · p. 17
  • Noiseless VQE simulations on small molecules show PECT can help difficult high-parameter ansatze converge and may reduce circuit depth, two-qubit gate counts, and optimization runtime.The conclusion describes LDCA and similar redundant ansatze as especially difficult to converge with existing strategies.Source: §VI · p. 17

APPENDIX

The appendices detail LDCA's gate structure, report resource-reduction tables for accurate H2O calculations, and examine parameter movement and potential-energy-surface behavior across ansatze.

Appendix B: Ansatz cost reductions for H2O

  • Tables II–IV report PECT's circuit-depth and two-qubit-gate reductions for UCCSD, 4-UpCCGSD, and 5-UpCCGSD at H2O bond lengths where both methods achieve errors below chemical accuracy.The supplied passages give the tables' scope but not their numerical cell values.Source: Appendix B · p. 20, Table II · p. 21, Table III · p. 22

Appendix C: Parameter dynamics of k-UpCCGSD and LDCA optimizations

  • Across the k-UpCCGSD and LDCA examples, earlier layers generally move less toward their final parameter values, but the ordering becomes less structured after some layers.LDCA for H2O is an exception: layer 6 changes less than layers 3–5.Source: Appendix C · pp. 20–21, Figure 11 · p. 23

Appendix A: LDCA Details

  • Each LDCA entangling block uses five two-qubit gates and three free parameters; shared parameter pairings preserve particle number when initialized, for example, from Hartree-Fock.Figure 10 shows an eight-qubit circuit and the block composition.Source: Appendix A · p. 20, Figure 10 · p. 21

Appendix D: Estimating the potential energy surface for H2O

  • For H2O, PECT yields a smoother UCCSD potential-energy surface than non-PECT, k-UpCCGSD surfaces are less smooth, and PECT-LDCA is described as smooth and accurate relative to the other methods.The UCCSD non-PECT surface has a bump at O-H = 2.4 Å.Source: Appendix D · pp. 21–22, Figure 12 · p. 24

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