PaperlayerView exact revision on arXiv

Source-linked AI summary

Gate-free state preparation for fast variational quantum eigensolver simulations: ctrl-VQE

Oinam Romesh Meitei, Bryan T. Gard, George S. Barron, David P. Pappas, Sophia E. Economou, Edwin Barnes, Nicholas J. Mayhall

arXiv:2008.04302v3quant-phcond-mat.str-elphysics.chem-ph

TL;DR

Near-term VQE is constrained by deep circuits and limited coherence, which hinder accurate simulations of strongly correlated systems. ctrl-VQE replaces circuit-based state preparation with optimized control pulses, and numerical simulations report accurate H2 and HeH+ dissociation curves. The work also applies the approach to a single LiH energy calculation.

  • Problem

    Deep VQE circuits face limited coherence and frequent gate errors, constraining accurate simulations of systems with significant entanglement.

    Source: §I · pp. 1–2

  • Method

    ctrl-VQE replaces the parameterized state-preparation circuit with optimized laboratory-frame control pulses while retaining molecular energy as the variational objective.

    Source: B. Control Variational Quantum… · p. 3

  • Results

    The maximum difference from FCI energy was 0.03 mHa for both H2 and HeH+, with average error 0.002 mH in both cases.

    Source: A. Dissociation of diatomic molecules · p. 6

  • Takeaways & Limitations

    An adaptive pulse-parameterization scheme identifies the minimal number of parameters needed for target accuracy, simplifying experimental implementation and avoiding over-parameterization.

    Source: §V · p. 12

    Classical simulations are computationally more expensive than typical VQE emulations, limiting the systems studied so far to very small ones.

    Source: §V · p. 12

Abstract

from arXiv · show

The variational quantum eigensolver (VQE) is currently the flagship algorithm for solving electronic structure problems on near-term quantum computers. This hybrid quantum/classical algorithm involves implementing a sequence of parameterized gates on quantum hardware to generate a target quantum state, and then measuring the expectation value of the molecular Hamiltonian. Due to finite coherence times and frequent gate errors, the number of gates that can be implemented remains limited on current quantum devices, preventing accurate applications to systems with significant entanglement, such as strongly correlated molecules. In this work, we propose an alternative algorithm (which we refer to as ctrl-VQE) where the quantum circuit used for state preparation is removed entirely and replaced by a quantum control routine which variationally shapes a pulse to drive the initial Hartree-Fock state to the full CI target state. As with VQE, the objective function optimized is the expectation value of the qubit-mapped molecular Hamiltonian. However, by removing the quantum circuit, the coherence times required for state preparation can be drastically reduced by directly optimizing the pulses. We demonstrate the potential of this method numerically by directly optimizing pulse shapes which accurately model the dissociation curves of the hydrogen molecule (covalent bond) and helium hydride ion (ionic bond), and we compute the single point energy for LiH with four transmons.

I. INTRODUCTION

The introduction presents VQE as a route to molecular simulation but identifies deep, error-prone circuits as a constraint on near-term devices. It introduces ctrl-VQE, which replaces the state-preparation circuit with direct optimization of laboratory-frame control settings.

I. INTRODUCTION

  • Approximate classical methods can be inaccurate for strongly correlated systems, while DMRG and selected configuration interaction assume correlations are low-dimensional or limited in scale.The introduction states that no polynomially scaling classical algorithm currently solves arbitrary molecular ground states.Source: §I · p. 1
  • Representing an exact, generally entangled molecular wavefunction in a classical product-state basis requires an exponentially large coefficient vector.The paper identifies wavefunction dimensionality and storage as sources of exponential classical modeling cost.Source: §I · p. 1
  • Quantum computing offers a way to represent the molecular wavefunction directly as a QPU state and measure molecular observables on that state.The paper describes the molecular and QPU state spaces as having identical Hilbert-space dimensions.Source: §I · p. 1

A. Variational Quantum Eigensolver

  • VQE variationally minimizes molecular energy by optimizing tunable gates in a parameterized circuit that defines the reachable state subspace.A Hartree–Fock reference and an ansatz provide the starting state and variational flexibility.Source: §I · p. 1, A. Variational Quantum Eigensolver · p. 2
  • Deeper circuits and frequent gate errors constrain near-term VQE accuracy, while increasing ansatz parameters also makes optimization more difficult.VQE repeatedly prepares the circuit and measures Hamiltonian operators to estimate energy, then updates parameters until convergence.Source: §I · pp. 1–2, A. Variational Quantum Eigensolver · p. 2

I. INTRODUCTION

  • ctrl-VQE replaces VQE’s parameterized state-preparation circuit with directly optimized laboratory-frame control settings, a strategy the authors describe as more feasible for near-term NISQ devices.The paper frames direct control as an alternative to improving error mitigation or increasing the ratio of coherence time to gate time.Source: §I · p. 2

B. Control Variational Quantum Eigensolver: ctrl-VQE

ctrl-VQE replaces VQE’s gate-based state-preparation circuit with a variational laboratory-frame pulse and minimizes the qubit-mapped molecular-energy objective. Starting from an HF state, it alternates quantum measurements with classical pulse updates on a specified hardware platform.

B. Control Variational Quantum Eigensolver: ctrl-VQE

  • ctrl-VQE minimizes the qubit-mapped molecular Hamiltonian’s expectation value by measuring the prepared state and classically updating pulse parameters until convergence.The procedure starts from an initial state such as HF; when the pulse representation can specify the target Hilbert space, the optimum is the minimum-energy state.Source: B. Control Variational Quantum… · p. 3
  • Total pulse duration is fixed during optimization and treated as a hyperparameter, although square-pulse segment times are also optimized.The reported optimal pulse shape is therefore optimal for a given total duration unless otherwise stated.Source: B. Control Variational Quantum… · p. 3
  • Unlike universal quantum algorithms, ctrl-VQE simulations require a specified physical platform, modeled here with coupled transmons.The device Hamiltonian assumes a closed linear chain with always-on direct couplings.Source: B. Control Variational Quantum… · p. 3
  • Interqubit couplings create an entangling control Hamiltonian in the interacting frame, and the paper identifies coupling strength g with electron correlation.The drives are single-qubit terms, while the interacting-frame ansatz evolves the reference state under the variational controls.Source: B. Control Variational Quantum… · p. 3
  • The device-and-control ansatz can compile any digital quantum circuit and has no intrinsic accuracy limit beyond quantum speed limits.Its added flexibility can make optimization harder, motivating simple pulse constraints.Source: B. Control Variational Quantum… · p. 4
  • Square pulses use piecewise-constant amplitudes, while Gaussian pulses are an alternative; amplitudes and drive frequencies are bounded in the simulations.For N transmons with n square-pulse segments per transmon, the optimization uses 2Nn parameters and L-BFGS-B.Source: B. Control Variational Quantum… · p. 4

III. COMPUTATIONAL DETAILS

The computational study uses a locally developed simulation program, standard quantum-software tools for checks, and PySCF molecular integrals in an STO-3G basis. It focuses on simple pulse waveforms while supporting analytic energy gradients for arbitrary pulse shapes.

III. COMPUTATIONAL DETAILS

  • Simulations used CtrlQ, checked against Qiskit and Qutip, with PySCF integrals and an STO-3G basis across two- or four-transmon devices.Device parameters are listed in Table I; the SI compares a device with larger transmon detuning.Source: §III · p. 4
  • The simulations use parameters typical of current superconducting transmons, and the results are reported as qualitatively insensitive to qubit frequency difference.The authors provide an SI comparison with a device having larger detuning.Source: §III · p. 4
  • The study focuses on restricted, few-parameter pulse waveforms to simplify experimental implementation, while CtrlQ supports arbitrary shapes with analytic energy gradients.The gradients are with respect to pulse amplitudes.Source: §III · p. 4

IV. RESULTS AND DISCUSSION

The results section examines ctrl-VQE on molecular dissociation curves and pulse durations, alongside comparisons with related quantum algorithms.

IV. RESULTS AND DISCUSSION

  • The section examines whether ctrl-VQE reproduces FCI bond-dissociation curves for two small diatomic molecules with different electron-correlation profiles.It also analyzes pulse-time dependence and compares the approach with a representative gate-based circuit and related quantum algorithms.Source: §IV · pp. 4–5
  • Table I lists device parameters for a four-transmon system; the H2 and HeH+ simulations use parameters for transmons 1 and 2.All parameters are in 2π GHz.Source: Table I · p. 5
  • The H2 and HeH+ energy curves use ctrl-VQE square pulses with two time segments.The shortest pulse duration at each geometry is plotted for both molecules.Source: Figure 2 · p. 5, Figure 3 · p. 5

A. Dissociation of diatomic molecules

ctrl-VQE closely reproduces the FCI dissociation curves of H2 and HeH+, despite their contrasting correlation behavior as bonds stretch.

A. Dissociation of diatomic molecules

  • As H2 stretches, its ground state becomes increasingly multiconfigurational, while the Hartree–Fock approximation becomes less accurate.The shrinking HOMO–LUMO gap accompanies the increasing multiconfigurational character.Source: A. Dissociation of diatomic molecules · p. 5
  • HeH+ becomes easier to model as its bond distance increases, and Hartree–Fock becomes exact at dissociation in the minimal basis used.The separated products are closed-shell He and H+; the passage notes this exactness is specific to the minimal basis.Source: A. Dissociation of diatomic molecules · p. 5
  • The shortest pulse duration at each geometry is plotted for both molecular dissociation curves.The plotted curves use square pulses to generate the molecular energies.Source: Figure 4 · p. 6
  • 0.03 mHa maximum FCI-energy difference and 0.002 mH average error were reported for both H2 and HeH+ across their dissociation curves.The resulting states have 99% overlap with the exact FCI ground states in all cases; stretched H2 can converge to a singlet–triplet superposition near degeneracy.Source: A. Dissociation of diatomic molecules · pp. 5–6

B. Effect of electron correlation on pulse duration

Pulse duration varies in opposite directions along the two dissociation curves: it increases for H2 and decreases for HeH+.

B. Effect of electron correlation on pulse duration

  • Pulse duration increases as H2 stretches but decreases for HeH+, matching their contrasting entanglement demands.The paper relates stronger correlation in stretched H2 to greater entanglement, while HeH+ becomes easier to model at longer distances.Source: B. Effect of electron correlation on… · p. 6
  • 1.0 ns pulses suffice for HeH+ at far bond distances, where its Hartree–Fock state is a good approximation to the FCI ground state.Above 2.0 Å, the initial and target states become degenerate, requiring only a slight modification of the initial state.Source: B. Effect of electron correlation on… · p. 6
  • At H2 dissociation, pulse duration is significantly longer than near equilibrium, but it does not increase monotonically with the Hartree–Fock energy.The authors suggest pulse duration reflects the different dynamics along the two dissociation curves.Source: B. Effect of electron correlation on… · p. 6

C. Leakage to higher energy states

Energy evolution can be nonmonotonic, and optimized pulses differ in leakage; the authors discuss the resulting shot-count trade-off.

C. Leakage to higher energy states

  • For each tested pulse duration, molecular energy initially rises before decreasing or oscillating and then rapidly converging to the exact energy.The figure plots energy difference from FCI above and leakage below over evolution time for H2 at a bond distance of 0.75 Å.Source: C. Leakage to higher energy states · p. 6, Figure 5 · p. 7
  • Some optimized pulses produce more leakage than others, which can increase the shots needed for precise expectation values because leaked shots may be discarded or used for normalization.The results include distinct optimized solutions for the same T = 29 ns.Source: C. Leakage to higher energy states · p. 6, Figure 5 · p. 7
  • With 10% leakage, the authors anticipate a commensurate increase in shot count for post-selection and do not expect leakage to be a fundamental limitation.They also report comparable SI results using an unnormalized-energy objective, which penalizes leakage.Source: C. Leakage to higher energy states · p. 7

D. Noise analysis: Imprecise control

The authors test how uncertainty in optimized ctrl-VQE drive settings affects H2 energy accuracy. Their simulations find tolerance to moderate parameter noise and no significant objective change when decoherence effects are included.

D. Noise analysis: Imprecise control

  • For each set of optimal ctrl-VQE drive parameters, 100 Gaussian-noisy parameter samples produce energy samples that are averaged to calculate the energy difference relative to the target ground-state energy.With this noise model, accuracy below a 10^-4 energy error remains achievable; energy differences become significant only when noise exceeds σ = 0.01.Source: D. Noise analysis: Imprecise control · p. 7
  • Only noise above σ = 0.01 makes ctrl-VQE energy differences significant; imprecise settings can still achieve energy error below 10^-4.The authors associate σ = 0.01 with errors of 10 MHz or 1 ns and report that these errors are of the same order as the parameters themselves.Source: D. Noise analysis: Imprecise control · p. 7
  • Lindblad simulations including T1 and dephasing effects did not significantly change the objective, which the authors attribute to the short pulse timescales.The analysis otherwise omits explicit finite-decoherence and dephasing noise.Source: D. Noise analysis: Imprecise control · p. 7

E. Comparison with Circuit Compilation Techniques

Circuit-compilation approaches shorten gate-based state preparation by optimizing analog pulses, but they retain a target circuit and can add compilation latency. ctrl-VQE instead optimizes pulses for a target state without specifying a circuit or unitary.

E. Comparison with Circuit Compilation Techniques

  • Circuit-based ansätze can require too many gates for current hardware beyond the smallest molecules, motivating efforts to reduce state-preparation time and coherence demands.The compilation techniques discussed retain a circuit as their target, unlike ctrl-VQE.Source: E. Comparison with Circuit Compilation… · pp. 7–9
  • GRAPE-based compilation can provide up to 5-fold speedups over standard gate-based compilation, but its computational cost and compilation latency make it impractical for iterative VQE, and it is limited in circuit size.The section also describes partial compilation as a way to combine fast gate-based compilation with pulse speedups.Source: E. Comparison with Circuit Compilation… · p. 8
  • Strict partial compilation precomputes pulses for fixed subcircuits, while flexible partial compilation optimizes low-depth blocks with one variational parameter each.Flexible partial compilation reduces latency around 80x and achieves greater pulse speedups than strict partial compilation, but remains slower to compile than standard gate-based methods.Source: E. Comparison with Circuit Compilation… · p. 8
  • Unlike compilation, ctrl-VQE does not target a circuit or a specific unitary; it optimizes pulse control to prepare one state.Many unitaries can minimize its objective because their behavior outside the initial and final states is unconstrained.Source: E. Comparison with Circuit Compilation… · p. 9

F. Decompiled control pulses

The decompiled-pulse analysis benchmarks ctrl-VQE pulse duration against equivalent gate-based preparation. The adaptive parameterization study then grows pulse segments as needed, finding molecule-dependent accuracy and a limit for LiH at the tested duration.

F. Decompiled control pulses

  • Decompiling the pulse evolution into a circuit provides a benchmark of gate-based state-preparation time against the pulse duration, although ctrl-VQE itself requires no state-preparation circuit.The study uses KAK decomposition of the evolution operator and, separately, constructs and transpiles a circuit for the resulting state vector.Source: F. Decompiled control pulses · p. 9
  • For H2 at 0.75 Å, ctrl-VQE prepares the state in 9 ns, versus 825 ns for a transpiled circuit and 1202 ns for a KAK-decomposed circuit.The two circuit durations estimate the time to execute decompiled state-preparation circuits on the mock IBMQ hardware.Source: F. Decompiled control pulses · p. 9

G. Adaptive update of pulse parameterization

  • Adaptive refinement starts with one square-pulse segment, then repeatedly splits a segment and reoptimizes from the preceding pulse shape until energy convergence.The drive frequencies are also optimized; segment switching times remain fixed after division.Source: G. Adaptive update of pulse… · pp. 9–10
  • H2 and HeH+ reach chemical accuracy with one segment, while LiH requires three; energy converges with two segments for H2 and HeH+ and five for LiH.The optimized total pulse durations are 9 ns for H2 and HeH+ and 40 ns for LiH.Source: G. Adaptive update of pulse… · p. 10
  • More than 100 additional LiH optimizations did not exceed the adaptive pulse's accuracy, suggesting a limit for LiH with a 40 ns pulse.The additional runs used a 10-segment square pulse, retaining switching times while randomizing amplitudes and drive frequencies.Source: G. Adaptive update of pulse… · p. 10

H. Comparison to gate-based ans¨atze

The authors compare ctrl-VQE pulse durations with circuit-based RY and UCCSD ansätze using IBMQ device calibration data. The circuit timings are substantially longer in the reported cases, with LiH UCCSD approaching the device's decoherence-time range.

H. Comparison to gate-based ans¨atze

  • For LiH, ctrl-VQE takes 40 ns, compared with 3485 ns for RY and 82169 ns for UCCSD; the UCCSD duration is already reaching the device decoherence-time limit.For H2 and HeH+, ctrl-VQE takes 9 ns, versus 519 ns for RY and 825 ns for UCCSD.Source: H. Comparison to gate-based ans¨atze · p. 11
  • For LiH, RY uses 9 CNOTs and 16 parameters, whereas UCCSD uses 245 CNOTs and 8 parameters.For H2 and HeH+, RY uses 1 CNOT and 4 parameters, while UCCSD uses 2 CNOTs and 3 parameters.Source: H. Comparison to gate-based ans¨atze · p. 11
  • The timing comparison uses IBMQ calibration data for gate-based circuits, while ctrl-VQE pulse times come from a simulated device with somewhat different parameters.The authors expect the pulse-time results to be close to those from direct calculations on IBMQ devices.Source: H. Comparison to gate-based ans¨atze · p. 11

V. SUMMARY AND FUTURE OUTLOOK

ctrl-VQE replaces state-preparation circuits with optimized hardware-level pulses. Numerical results cover H2 and HeH+ dissociation and a LiH single-point energy, while classical simulations remain limited to very small systems.

V. SUMMARY AND FUTURE OUTLOOK

  • 0.02 kcal/mol maximum error from exact FCI energy—within chemical accuracy—across H2 and HeH+ bond dissociation.The reported accuracy holds throughout the dissociation curves.Source: §V · pp. 11–12
  • ctrl-VQE replaces state-preparation circuits with optimized pulse shapes, using adaptive parameter growth to find the minimum pulse parameterization for target accuracy.The adaptive scheme is intended to simplify experimental implementation and avoid over-parameterization.Source: §V · pp. 11–12
  • At a single bond distance, the authors also compute LiH energy; modest pulse-parameter counts achieve energy convergence as problem size increases.Source: §V · p. 12
  • Short pulse durations minimize decoherence and dephasing losses and provide significant state-preparation speedups versus standard gate-based ansätze.The authors describe this as a step toward more accurate VQE simulations on larger, strongly correlated systems.Source: §V · p. 12
  • Classical ctrl-VQE simulations are more computationally expensive than typical VQE emulations, so the systems studied so far are very small.The authors identify larger-system studies and more sophisticated pulse-shape constraints as future work.Source: §V · p. 12
Loading 2008.04302v3…