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Quantum optimization using variational algorithms on near-term quantum devices

Nikolaj Moll, Panagiotis Barkoutsos, Lev S. Bishop, Jerry M. Chow, Andrew Cross, Daniel J. Egger, Stefan Filipp, Andreas Fuhrer, Jay M. Gambetta, Marc Ganzhorn, Abhinav Kandala, Antonio Mezzacapo, Peter Müller, Walter Riess, Gian Salis, John Smolin, Ivano Tavernelli, Kristan Temme

arXiv:1710.01022v2quant-ph

TL;DR

Near-term quantum processors lack the scale and error resilience of universal fault-tolerant machines, motivating methods that can use limited qubits and circuit depth. The paper analyzes VQE and quantum volume for optimization and chemistry, including fermion-to-qubit mappings, trial wavefunctions, and error mitigation. It reports promising small-scale optimization and chemistry demonstrations while emphasizing that reduced effective error, short circuits, and further mitigation development remain necessary.

  • Problem

    Near-term devices offer only limited qubits, error correction, coherence, and connectivity, creating a challenge for useful optimization and quantum-chemistry calculations.

  • Method

    The paper develops a VQE-based hybrid quantum-classical framework, analyzes quantum volume, maps fermions to qubits, and considers coupled-cluster, heuristic, and error-mitigation techniques.

  • Results

    The MaxCut solution was found with probability higher than 95% after 100 VQE trial steps in an ideal quantum simulation.

  • Takeaways & Limitations

    Near-term quantum computation may address useful chemistry, materials, and classical-optimization problems if circuit depth, effective errors, and mitigation are improved.

  • Takeaways & Limitations

    VQE is not resilient against decoherence and gate errors, while coupled-cluster implementations for large molecules require further truncation because of limited circuit depth.

Abstract

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Universal fault-tolerant quantum computers will require error-free execution of long sequences of quantum gate operations, which is expected to involve millions of physical qubits. Before the full power of such machines will be available, near-term quantum devices will provide several hundred qubits and limited error correction. Still, there is a realistic prospect to run useful algorithms within the limited circuit depth of such devices. Particularly promising are optimization algorithms that follow a hybrid approach: the aim is to steer a highly entangled state on a quantum system to a target state that minimizes a cost function via variation of some gate parameters. This variational approach can be used both for classical optimization problems as well as for problems in quantum chemistry. The challenge is to converge to the target state given the limited coherence time and connectivity of the qubits. In this context, the quantum volume as a metric to compare the power of near-term quantum devices is discussed. With focus on chemistry applications, a general description of variational algorithms is provided and the mapping from fermions to qubits is explained. Coupled-cluster and heuristic trial wave-functions are considered for efficiently finding molecular ground states. Furthermore, simple error-mitigation schemes are introduced that could improve the accuracy of determining ground-state energies. Advancing these techniques may lead to near-term demonstrations of useful quantum computation with systems containing several hundred qubits.

1. Introduction

Near-term quantum devices may support useful optimization despite limited qubit counts, error correction, coherence, and connectivity. The paper presents VQE as a hybrid strategy for classical and quantum-chemistry optimization, while identifying fermion-to-qubit mappings and errors as practical challenges.

  • Motivation: Near-term devices with several hundred physical qubits may enable useful calculations before universal fault-tolerant quantum computers are available.Universal fault-tolerant systems are expected to require millions of physical qubits and high-fidelity gates.
  • Motivation: Optimization problems become difficult classically because required computational resources can scale exponentially with problem size.The paper distinguishes molecular ground-state and dynamics problems from classical problems mapped to Ising-type Hamiltonians.
  • Quantum chemistry challenges: Fermion-to-qubit representations require strings of qubit operators, creating long-range correlations and demanding connectivity and gate counts.Digital simulation also requires decomposition into discrete time steps and consequently long gate sequences.
  • Variational quantum eigensolver: VQE uses short-depth quantum circuits to prepare parameterized trial states while a classical computer optimizes gate parameters from measured expectation values.The approach generates high-dimensional trial wavefunctions with single-qubit and entangling gates on the quantum processor.
  • Practical limitations: Decoherence and gate errors can make expectation-value estimates inaccurate, while available error correction requires substantial qubit overhead.The paper therefore motivates further development of error-mitigation schemes that use no or few additional ancilla or code qubits.

2. Quantum volume, a metric for near-term quantum devices

Quantum volume is proposed as an architecture-neutral measure combining qubit count and executable circuit depth. Its analysis shows that connectivity, gate overhead, errors, and software mapping jointly determine near-term computational capability.

  • Definition: Quantum volume evaluates a device using both the number of physical qubits N and the allowable circuit depth d.The metric is intended to reflect whether a device can run a given algorithm rather than optimize qubit count or depth separately.
  • Effective error rate: Effective error rate ϵeff incorporates gate errors, connectivity overhead, parallelism, and the available gate set.Limited connectivity can require additional SWAP gates, while hardware or compilation improvements can reduce effective overhead.
  • Scaling: The allowable depth scales approximately as d ≃1/(Nϵeff), linking simultaneous two-qubit errors to executable circuit depth.At ϵeff = 10^-4, the paper gives depth 10 on a 1000-qubit device and depth 100 on a 100-qubit device.
  • Scaling: Effective error rate can depend on qubit number through system complexity and crosstalk, and on the sophistication of hardware-aware scheduling.Consequently, both hardware and software improvements affect ϵeff(N).
  • Scaling: At constant ϵeff, increasing the number of qubits can reduce allowable depth, so selecting a well-connected subset may improve quantum volume.The paper notes that a larger machine is not automatically more useful when additional qubits have the same fidelity.
  • Implications: Quantum volume is architecture-neutral and supports fair comparison of systems with different hardware characteristics, but increasing qubit count improves power only alongside lower effective error.The figure analysis identifies a tipping point where d = 1/(Nϵeff) = N and typical current effective error rates exceed 10^-3.

3. Exploring Hilbert space with the variational quantum eigensolver

The variational quantum eigensolver uses a hybrid quantum-classical loop to prepare parameterized trial states, measure a Hamiltonian cost function, and update parameters toward the ground state. Its implementation combines Pauli-string measurements on the quantum processor with classical optimization, while minimizing hardware queries.

  • Near-term quantum devices motivate hybrid algorithms that prepare multiqubit states quantum mechanically while optimizing control parameters classically.These architectures target hardware with limited quantum volume and without full error correction.
  • VQE method: The cost function Eq(θ) is obtained from the expectation value of a qubit Hamiltonian Hq, which can represent molecular, condensed-matter, or classical optimization problems.The Hamiltonian must map to interacting qubits with a non-exponentially increasing number of terms.
  • VQE method: VQE generates a trial state |Ψ(θ)⟩ using a gate sequence parameterized by control parameters θ.The gate set should efficiently explore Hilbert space and include states capable of representing the minimization solution.
  • Expectation-value measurement: Expectation values are measured by repeatedly sampling qubit populations, using prerotations for non-σz Pauli operators, and averaging products of measurement outcomes.The Hamiltonian expectation value is assembled by summing Pauli-string expectation values with coefficients hα.
  • Classical optimization loop: A classical optimizer minimizes Eq(θ) and feeds updated parameters back to the quantum processor for renewed trial-state preparation.Because each parameter set requires reprogramming the hardware, the algorithm should minimize quantum-processor queries.

4. Quantum chemistry with qubits

Quantum-chemistry VQE requires mapping fermionic Hamiltonians onto qubits and preparing expressive trial states within hardware constraints. The paper compares coupled-cluster and hardware-efficient constructions, illustrating how entangling depth affects molecular-energy accuracy.

  • 4.1. Mapping fermions to qubits: Second-quantized electronic Hamiltonians must be mapped from fermionic operators to qubit operators because fermionic statistics prevent direct implementation on qubit processors.The mapping introduces parity strings and long-range correlations, increasing connectivity and gate requirements.
  • 4.1. Mapping fermions to qubits: The Jordan-Wigner mapping stores occupation locally but uses a nonlocal O(N) parity function, whereas binary-tree mappings scale as O(log(N)).Fermionic-symmetry tapering can further reduce the qubit simulation space.
  • 4.2. Coupled cluster trial wavefunctions: Full configuration interaction scales factorially with electron number, motivating coupled-cluster trial states that systematically sample relevant excitations up to a chosen degree.Unitary coupled-cluster states use an exponentiated cluster operator acting on a reference Slater determinant, with parameters optimized by VQE.
  • 4.2. Coupled cluster trial wavefunctions: UCC expansions are commonly truncated at double or triple excitations, but large systems still require further truncation because current devices have limited circuit depth.The number of UCCSD coefficients grows as the fourth power of the number of orbitals, and truncation effects on accuracy require further study.
  • 4.3. Hardware-efficient trial states suitable for near-term quantum hardware: Hardware-efficient trial states alternate single-qubit Euler rotations with entangling operations to generate highly entangled states suited to available connectivity.The heuristic construction applies D entanglers interleaved with rotations to an initial computational-basis state.
  • 4.4. Small molecules calculated with the variational quantum eigensolver: Two or more entanglers reproduce the exact hydrogen energy on an ideal simulator, while LiH and BeH2 require D = 8 and D = 28, respectively, for chemical accuracy without noise.Decoherence and finite sampling reduce the practical optimal depth for current hardware to between zero and two entanglers.

5. Classical optimization with qubits

The paper maps classical optimization problems to Ising-type Hamiltonians and uses variational circuits to search for low-cost states. QAOA and hardware-efficient heuristic trial states provide short-depth approaches, with a four-node MaxCut simulation finding the solution with probability above 95%.

  • 5.1. Quantum approximate optimization algorithm with short depth: QAOA alternates evolution under a cost Hamiltonian and a mixing Hamiltonian, with circuit level D controlling algorithmic complexity.The cost Hamiltonian encodes the binary objective, while the mixer guides exploration toward its ground state.
  • 5.1. Quantum approximate optimization algorithm with short depth: Variationally optimized parameters can reach the target ground state with high accuracy at relatively small circuit depth.Fixed interpolation approximates adiabatic evolution, whereas VQE-selected parameters can provide a more direct path.
  • 5.2. Variational quantum eigensolver applied to the MaxCut problem: MaxCut seeks a bipartition maximizing the total weight of edges crossing between the two subsets.The problem is NP-complete and can be represented through binary variables and an equivalent Ising Hamiltonian.
  • 5.2. Variational quantum eigensolver applied to the MaxCut problem: A hardware-efficient heuristic trial state uses entangling gates and single-qubit rotations to search the quantum-state space with real amplitudes.The number of entanglers defines the circuit level, while the parameterization enables a compact search over states.
  • 5.2. Variational quantum eigensolver applied to the MaxCut problem: 95%: the four-node MaxCut solution probability after 100 VQE trial steps on an ideal quantum simulator.The calculation used heuristic trial states and measured pairwise σz correlators.

6. Classical robust optimizers for measured expectation values

VQE optimization must cope with local minima, stochastic measurement noise, and substantial sampling overhead. The paper discusses robust classical optimizers, commuting-group measurements, and hardware improvements to make feedback cycles more practical.

  • Local minima: VQE optimization can become trapped in a local minimum corresponding to an excited state rather than the ground state.Suggested remedies include simulated-annealing steps, multiple starting points, and hybrid greedy-Powell searches.
  • Noisy expectation values: O(1/√s): the energy-estimation error from s Hamiltonian-term samples decreases with the square root of the sample count.Grouping commuting Pauli operators reduces separate measurements and improves sampling statistics.
  • Noisy expectation values: Optimizer choices must tolerate stochastic cost-function fluctuations caused by finite quantum sampling.Analytical variational circuits may support gradients, but noisy measurements require methods suited to stochastic feedback.
  • Optimization overhead: SPSA offers constant variational-parameter overhead and robustness to stochastic fluctuations, addressing repeated function-evaluation costs.It has been used for molecular structure problems.
  • Optimization overhead: Integrated active reset can accelerate VQE execution by shortening repeated initialization, gate-application, and measurement cycles.Accurate cost measurement requires many such repeated cycles on the quantum processor.

7. Prospects of fighting decoherence without full error correction

Decoherence and gate noise limit useful near-term quantum computations, even for short-depth molecular simulations. The paper presents zero-noise extrapolation as an error-mitigation strategy that needs no extra ancilla or code qubits and can substantially improve expectation estimates.

  • Noise constraints: Near-term quantum computations are constrained by environmental coupling, which limits both their usable duration and size.These limits must be balanced against the scaling advantages expected from quantum algorithms.
  • Noise constraints: Short-depth small-molecule simulations already show decoherence effects that must be mitigated for the simulations to be useful.The paper motivates error mitigation as an alternative to full error correction for near-term devices.
  • Zero-noise extrapolation: Expectation-value estimation after a short-depth circuit is a central task for many applications, and its accuracy can be improved with modest time overhead.The described approach requires no fresh ancilla or code qubits.
  • Zero-noise extrapolation: Richardson extrapolation estimates the noise-free expectation value by measuring the observable at several rescaled noise rates and extrapolating to zero noise.The method assumes weak, time-independent noise and uses rescaled dynamics to emulate different effective noise strengths.
  • Zero-noise extrapolation: 10^-6 to 10^-11: reported relative-error range for noise-free expectation values under the rescaling-based mitigation procedure.The estimate cancels leading error terms through the chosen extrapolation order.

8. Conclusion

The conclusion identifies quantum volume, error reduction, compact trial states, efficient mappings, circuit optimization, and robust classical feedback as prerequisites for useful near-term computation. It emphasizes quantum chemistry as a leading application while recognizing hardware and algorithmic limits.

  • Processor capability: Quantum volume should assess processor capability by combining qubit count with the circuit depth the device can reliably execute.The metric is proposed to compare different prototypes on a fair basis.
  • Processor capability: 0.01%: estimated effective error rate required for a depth-100 algorithm on a 100-qubit device.The estimate reinforces the need for short-depth algorithms and low-overhead error mitigation.
  • Prospects: Quantum chemistry may offer stronger advantages than Ising-type optimization because chemistry Hamiltonians contain non-stoquastic terms that are difficult to solve exactly classically.For Ising-type problems, the amount of quantum speedup remains unclear because many fast classical algorithms exist.
  • Remaining challenges: Practical VQE requires improved coherence and control, hardware-efficient or problem-specific trial states, efficient fermion-to-qubit mappings, circuit optimization, and suitable classical optimizers.These requirements address gate errors, trial-state expressivity, connectivity, circuit depth, and noisy feedback.
  • Prospects: Near-term devices with hundreds of qubits and limited coherence times have several promising approaches available, with applications expected in chemistry, materials science, and classical optimization.The conclusion frames these applications as contingent on overcoming remaining challenges.
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