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Towards Practical Quantum Variational Algorithms

D. Wecker, M. B. Hastings, M. Troyer

arXiv:1507.08969v2quant-phcond-mat.str-el

TL;DR

Short-circuit quantum state preparation is promising for near-term quantum computers, but practical variational methods need accurate, optimizable ansätze and manageable energy estimation. The paper proposes Hamiltonian variational states motivated by adiabatic preparation, tests them on Hubbard ladders and molecules, and finds strong accuracy with a practical scope mainly for Hubbard-like models.

  • Problem

    Practical variational state preparation needs short circuits that can represent accurate ground states while keeping optimization and energy-measurement costs manageable.

  • Method

    The paper constructs Hamiltonian variational states from sequential rotations by Hamiltonian terms, assigning arbitrary variational angles rather than annealing-derived Trotter angles.

  • Results

    The Hamiltonian variational approach obtains higher overlap than UCC-based ansätze for larger, more strongly interacting systems, while requiring fewer evaluations for good parameters.

  • Takeaways & Limitations

    The approach is potentially practical for Hubbard models and related systems, where fewer Hamiltonian terms, limited energy ranges, and translation invariance reduce measurement demands.

  • Takeaways & Limitations

    For quantum-chemistry applications to molecules, the number of measurements needed for sufficiently accurate energy estimates is astronomical, making the variational algorithm impractical in its current form.

Abstract

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The preparation of quantum states using short quantum circuits is one of the most promising near-term applications of small quantum computers, especially if the circuit is short enough and the fidelity of gates high enough that it can be executed without quantum error correction. Such quantum state preparation can be used in variational approaches, optimizing parameters in the circuit to minimize the energy of the constructed quantum state for a given problem Hamiltonian. For this purpose we propose a simple-to-implement class of quantum states motivated by adiabatic state preparation. We test its accuracy and determine the required circuit depth for a Hubbard model on ladders with up to 12 sites (24 spin-orbitals), and for small molecules. We find that this ansatz converges faster than previously proposed schemes based on unitary coupled clusters. While the required number of measurements is astronomically large for quantum chemistry applications to molecules, applying the variational approach to the Hubbard model (and related models) is found to be far less demanding and potentially practical on small quantum computers. We also discuss another application of quantum state preparation using short quantum circuits, to prepare trial ground states of models faster than using adiabatic state preparation.

I. INTRODUCTION

The paper introduces Hamiltonian variational states: short circuits composed of Hamiltonian-term rotations with freely optimized angles, motivated by adiabatic preparation. It targets accurate state preparation with modest depth and parameter counts, while addressing optimization and measurement costs.

  • I. INTRODUCTION: The method uses modest gate depth and far fewer variational parameters than system-size-scaled alternatives such as UCC.For four-fermion-truncated UCC, the parameter count scales as the fourth power of the number of orbitals at constant filling fraction.
  • I. INTRODUCTION: The analysis tests Hamiltonian variational states on Hubbard models up to 12 sites, equivalently 24 spin-orbitals, including difficult highly degenerate non-interacting cases.The authors also compare the approach with UCC and Rxx on quantum-chemistry systems.
  • I. INTRODUCTION: For the Hubbard model, the Hamiltonian variational method is well suited to the small number and simple form of its interaction terms.The corresponding UCC terms produce larger and more complicated circuits for this model.
  • I. INTRODUCTION: Hamiltonian variational states use rotations generated by Hamiltonian terms, with arbitrary variational angles instead of angles fixed by annealing Trotterization.This flexibility permits shorter rotation sequences.
  • I. INTRODUCTION: The ansatz requires optimization that avoids false minima, and quantum energy estimation can require many samples because sampling error decreases only as the inverse square root of sample count.Phase estimation offers another energy-estimation route but remains probabilistic and still requires many runs to estimate average energy.
  • I. INTRODUCTION: Once optimized, the classical variational parameters can quickly recreate the state for repeated measurements, making short-circuit preparation useful beyond finding a single energy estimate.The paper frames short-circuit preparation as attractive for classically intractable states without stringent coherence-time and gate-fidelity requirements.

II. THE HUBBARD MODEL

The study uses two-leg Hubbard ladders from 4 to 12 sites, with specified boundary conditions, filling, spin sectors, and size-dependent non-interacting degeneracies. These spectral and spin details determine the initial-state structure and affect problem difficulty.

  • II. THE HUBBARD MODEL: The Hubbard scaling study examines two-leg ladders with N = 4, 6, 8, 10, 12 sites, arranged as an N/2-by-2 square lattice.The models use periodic boundaries horizontally and open boundaries vertically.
  • II. THE HUBBARD MODEL: The Hamiltonian is decomposed as H = h_h + h_v + h_U, containing horizontal hopping, vertical hopping, and on-site repulsion terms.All horizontal and vertical bonds have the same strength in the stated ladder geometry.
  • II. THE HUBBARD MODEL: The systems are studied at half-filling with N electrons on N sites, using equal numbers of up- and down-spin electrons except where the correct spin sector requires otherwise.For N = 6 and 10, the selected sector has N/2 + 1 up electrons and N/2 − 1 down electrons.
  • II. THE HUBBARD MODEL: At U = 0, the ground-state degeneracy varies with N: N = 4, 6, 10 have degeneracies, N = 8 is unique, and N = 12 has a 36-fold degeneracy.These degeneracies affect the difficulty of solving the interacting problem and reduce Slater-determinant overlap with the true ground state.
  • II. THE HUBBARD MODEL: At U = 2, the ground state is a singlet for N = 4n and a triplet for N = 4n + 2, so the initial state is chosen in the correct spin sector.A small perturbation is used for N = 4 to select a unique ground state when needed.

C. The variational ansatz

The annealed variational method builds short Hamiltonian-based circuits and uses staged optimization to improve robustness and accuracy. For the Hubbard models studied, increasing steps produced strong overlaps, including 97.9% for the 12-site case with 45 parameters.

  • Ansatz construction: The ansatz uses repeating Hamiltonian-term rotations with variational angles, implemented through Trotterized unitary products.The construction approximates evolution under interpolating Hamiltonians while optimizing the rotation angles rather than fixing them by annealing.
  • Annealed optimization: The annealed method avoids local-minimum problems observed in direct optimization, where energy and ground-state overlap could vary substantially between runs.An energy decrease can coincide with lower ground-state overlap when excited-state amplitudes are redistributed.
  • Annealed optimization: Sequential optimization targets intermediate Hamiltonians before a final global search over all parameters.The method starts from the ground state of H0, advances through intermediate targets, and uses the resulting parameters to initialize full optimization.
  • Results: For S > 3, annealed optimization becomes more accurate as the number of steps increases, without the convergence issues of global optimization.For S = 3 and N > 4, it was marginally worse but required fewer energy evaluations, sometimes by a factor of five.
  • Results: 97.9% overlap is achieved for N = 12 with 45 parameters, while N = 10 reaches 93.7% overlap with only 9 parameters.The 12-site model and the π-flux model are distinctly more difficult, yet the reported overlaps remain high.

E. Inexact Optimization: Gate Count and Run Time

Inexact energy estimates make optimization statistically demanding because small parameter changes produce small energy differences. The reported Hubbard-model implementation nevertheless uses short individual circuits, although total sampling time can be substantial.

  • Sampling challenge: Small energy changes between nearby parameter points require many samples to determine whether optimization improved.This motivates modifying the optimization algorithm when sampling is limited.
  • Optimization with sampling: The stochastic coordinate-search procedure obtained more than 98% ground-state overlap using 506 point evaluations and 4.3 × 10^7 total samples.Each point used an average of 8.5·10^5 samples across ten runs.
  • Scaling: The total gate-count cost scales linearly with the number of steps S.The state-preparation rotations and Hamiltonian unitaries contribute to this scaling.
  • Gate cost: About 1000 gates are required for one S = 2, N = 8 run, including state preparation, unitary implementation, and measurement.The estimate assumes one run measures only one of four commuting term sets.
  • Runtime: Ignoring parallelization, the complete estimate is roughly 47 hours at a 1 µs gate time, despite moderate per-run gate requirements.The total time includes 4.3 × 10^7 samples, four commuting sets, and 1000 gates per run.

A. The electronic structure Hamiltonian

The quantum-chemistry formulation uses a general second-quantized electronic Hamiltonian in a Hartree–Fock orbital basis. The proposed ansatz groups its terms into diagonal, hopping, and exchange sectors and applies corresponding structured unitaries.

  • Hamiltonian: The electronic Hamiltonian includes general two-body interactions, with indices p, q, r, and s labeling spin-orbitals.The long-range Coulomb interaction gives the Hamiltonian its general form.
  • Hamiltonian: The calculations first use Hartree–Fock orbitals as an orthogonal basis, while classical simulation remains limited to very small basis molecules.The paper studies HeH+, H2O, BeH2, and artificial hydrogen chains.
  • Variational ansatz: The chemistry ansatz groups terms as H = Hdiag + Hhop + Hex, representing diagonal, hopping, and exchange contributions.The exchange sector contains the remaining exchange terms for distinct spin-orbital indices.
  • Variational ansatz: The initial state is the ground state of Hdiag, and Hhop annihilates it in the Hartree–Fock basis.Udiag is exact because its terms commute, while hopping and exchange unitaries are implemented with structured product approximations.
  • Hydrogen chains: For hydrogen chains, the ansatz separates exchange terms into occupancy-changing and remaining components, using four parameters.The two parts are denoted Ho and Hrest and are controlled separately.

C. Unitary Coupled Cluster Ansatz

The UCC comparison uses symmetry-compatible operator subsets, including RAA, ROO/UCC, RNO, and RAO, with real parameters in the reported runs. The number of retained terms can scale steeply, while molecule-specific results show accuracy and evaluation-cost trade-offs.

  • Parameter choices: Real parameters were used in all reported runs, because more accurate results were found than with imaginary parameters.General complex parameters could provide more flexibility but would further increase the parameter count.
  • Operator variants: RAA keeps all quadratic and quartic terms, ROO equals UCC, RNO keeps the fewest terms, and RAO keeps all quadratic but selected quartic terms.The selections depend on whether terms annihilate the Hartree-Fock state.
  • Molecule results: For small molecules such as HeH+ and H2O, all methods find the ground state to very high accuracy.The passage attributes this possibly to the large number of parameters relative to Hilbert-space dimension.
  • Term scaling: At half-filling, RAA has O(NSO^4) quartic terms, while RAO, RNO, and ROO retain roughly one-eighth of them.The reduction is a constant-factor gain at half-filling and becomes larger away from half-filling.
  • Molecule results: For BeH2, RAA reaches 0.157 mHa energy error after over 2 × 10^5 evaluations, whereas the Hamiltonian variational method uses 5000−10000 evaluations.RAA improves most rapidly up to roughly 2 × 10^4 evaluations, reaching roughly 0.5 mHa error.

D. Hydrogen chains

Hydrogen-chain simulations test scaling across N = 2 to N = 10 atoms and compare Hamiltonian variational states with Rxx variants. Hamiltonian variational methods obtain higher overlap at larger N, while UCC becomes less accurate and harder to optimize as size increases.

  • Hamiltonian variational results: At larger N, the Hamiltonian variational method obtains significantly higher overlap; four parameters improve performance, while three require larger S for comparable accuracy.The comparison is reported across the hydrogen-chain sizes studied.
  • Rxx comparisons: For H6, energy performance improves from RNO through ROO, RAO, and RAA as more Rxx terms are included, but H8 and H10 reverse this ordering for overlap.At a fixed evaluation count, RAA generally performs better, often by only a small amount.
  • Convergence: For H10, Rxx energy error is approximately 0.2 after roughly 5 × 10^4 evaluations, followed by much slower improvement.RNO reaches the cited later values only after over 2.5 × 10^5 evaluations.
  • Scaling comparison: As N increases, the annealed variational approach becomes increasingly better than UCC, which is less accurate and more difficult to optimize at larger sizes.UCC performs well at small sizes, potentially because its parameter count is large relative to the Hilbert-space dimension.
  • Extensions: TRxx combines multiple Hamiltonian-variational steps with separate parameters for every term, increasing flexibility but potentially making optimization difficult.For Hubbard systems, UCC requires substantially greater circuit depth because of its additional terms.

IV. RESOURCE ESTIMATES FOR PRACTICAL APPLICATIONS

The paper frames practical resource estimates as necessary for applications beyond current classical capabilities.

  • Practical scope: Practical applications require resource estimates for regimes that might go beyond what can currently be done classically.The section introduces a discussion of these resource requirements.

A. Hubbard model

For the Hubbard model, the paper estimates resources for larger lattices using parallelizable Hamiltonian-term circuits and measurement assumptions. A 100-site system requires about 200 qubits and roughly 600,000 samples under the stated parameters.

  • Qubit and depth requirements: Lattices beyond 10 × 10, with N ≥ 100 sites, require about 200 qubits, plus slightly more ancillas for circuit parallelization.Parallelizing Hamiltonian-term circuits should not substantially increase parallel circuit depth, apart from potentially requiring more steps S.
  • Measurement assumptions: Assuming hopping-term variance of order 1 and double-occupancy variance of order 1/U, the measurement error estimate uses these reduced-variance terms.The double-occupancy variance reduction is attributed to suppression of double occupancy to about t/U at large U.
  • Sample complexity: For t = 1, U = 8, N = 100, and five measured terms, the estimate is M ≈ 120,000 samples per term, or about 600,000 total samples.The estimate assumes N measurements can be performed in parallel in each run.

B. Quantum chemistry

Quantum-chemistry energy estimation requires many measurements, making the variational approach resource-intensive for molecules, especially larger systems. The estimates become impractical as Hamiltonian complexity and circuit costs increase.

  • Energy-error estimates allocate measurements across Hamiltonian terms according to coefficient magnitudes, with variances bounded by Var(O_i) ≤ 1.The Hamiltonian is written as a sum of terms h_iO_i, and the optimization uses M_i ∝ |h_i| measurements.
  • 108 to 10^9 samples are required for each energy evaluation of HeH+, BeH2, and H2O to achieve 1mHa accuracy.The reported sums of Hamiltonian coefficient magnitudes are 11.3Ha, 12.3Ha, and 36Ha, respectively.
  • The molecular sample requirement is about 1000 times larger than for the Hubbard model, which already required on the order of a few days to optimize.
  • Small-molecule simulations would require a massively parallel cluster of quantum computers because their circuits are more complex and measurements are costly.This estimate assumes 1µs gate and measurement times.
  • 10^13 samples per energy evaluation are required for Fe2S2 with 112 spin-orbitals, and roughly 10^19 samples would be needed across 10^6 evaluations.A single Hamiltonian-circuit sample is estimated to require approximately 2 × 10^8 gate executions.
  • Noisy optimization may still require many high-accuracy energy evaluations because individual search steps often yield only very small energy improvements.

V. DISCUSSION

The discussion presents Hamiltonian variational states as compact, accurate circuits with practical advantages for Hubbard models and state preparation. However, measurement costs remain a major obstacle for quantum chemistry, and optimization methods still involve unresolved trade-offs.

  • A modest number of parameters can achieve very large ground-state overlap while avoiding the impractical optimization over all circuits of a fixed depth.The circuits are selected from Hamiltonian terms rather than arbitrary unitary circuits.
  • For larger systems with stronger interactions, Hamiltonian variational states obtain significantly higher overlap than UCC and Rxx ansatz wave functions.
  • Quantum-chemistry measurement demands are astronomical and make the variational algorithm impractical in its current form, whereas Hubbard-model demands are less challenging.The Hubbard advantage is associated with fewer terms, a limited energy range, and translation invariance.
  • Small Trotter numbers suffice for UCC and Rxx, reducing the circuit depth required by those methods.
  • For N = 8, variational preparation reached 0.99 overlap using S = 3, compared with 8 second-order Trotter-Suzuki steps for the optimized anneal.
  • For N = 12, the variational method used S = 19, compared with 640 annealing steps for the optimized procedure.The reported annealing result reached 0.9883 overlap.
  • Short-circuit state preparation may support property measurement and annealing to larger states by preparing multiple copies of smaller ground states.
  • Parameter optimization treated energy as a black-box function, while derivative estimation would roughly double circuit depth and has unclear optimization benefits.
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