Variational quantum eigensolver: what it estimates and what is not proved

Definition

The variational quantum eigensolver, VQE, estimates the lowest eigenvalue of a Hamiltonian by preparing a parameterized quantum state and letting a classical optimizer choose the parameters. Peruzzo, McClean, and coauthors introduced it in 2014 and demonstrated it on a photonic processor, computing a ground-state energy for the helium hydride ion, HeH+.

It is a variational quantum algorithm. The quantum device prepares a state from an ansatz and estimates an expectation value. The classical device updates the parameters. There is no theorem, in the papers this library cites, that the loop reaches the ground state in time polynomial in the number of orbitals or qubits.

Intuition

The variational principle of quantum mechanics is the reason the number means something. For any state you can prepare, the expected energy is at least the ground-state energy. Searching a family of states for a low expectation cannot undershoot the true ground state. It can stop far above it if the family is too small, if the optimizer fails, or if the energy estimate is too noisy to follow.

The design goal in the 2014 paper is coherence, not asymptotics. Quantum phase estimation can read an eigenvalue of a prepared eigenstate, and Aspuru-Guzik, Dutoi, Love, and Head-Gordon had already proposed that route for molecular energies. Phase estimation asks for a long coherent evolution, which is an application of Hamiltonian simulation. VQE replaces that evolution with many short preparations. Each shot measures one term, or one group of terms, in a decomposition of the Hamiltonian. The classical optimizer sees a scalar and proposes the next parameters.

McClean, Romero, Babbush, and Aspuru-Guzik develop that theory in 2016. A Hermitian operator is written as a weighted sum of Pauli operators. The energy is the same weighted sum of Pauli expectations. Hamiltonian averaging estimates those expectations by repeated state preparation. The number of preparations depends on the variances of the terms and on the precision you ask for. They also discuss grouping commuting operators to reduce the number of distinct measurement settings. None of that analysis is a bound on the outer optimization.

Unitary coupled cluster is the structured ansatz they treat for chemistry: a parameterized unitary built from fermionic excitations, prepared by a Trotterized circuit. A hardware-efficient ansatz is the other family, used when the device suggests the gates more than the molecule does.

What has been run

Peruzzo and coauthors report a photonic demonstration on HeH+. Kandala and coauthors, in 2017, run a hardware-efficient VQE on a superconducting processor for the molecules LiH and BeH2 and for a four-qubit Heisenberg model. Those are real experiments on small systems. They are not calculations of reaction energies for the molecules industrial chemistry asks about, and the papers do not claim that they are.

The NISQ setting Preskill named is the informal home of this style of algorithm: devices large enough to be interesting and too noisy for the long phase-estimation circuits. VQE was written to fit that constraint. Fitting the constraint is not the same as an advantage over classical quantum-chemistry methods.

Complexity

Do not quote a big-O runtime for VQE as if it were Shor's algorithm. The inner estimate, given a state, is a sampling problem whose cost grows as the requested precision tightens and as the Hamiltonian gains terms. The outer loop is non-convex optimization over the ansatz parameters. McClean, Boixo, Smelyanskiy, Babbush, and Neven show a specific failure mode for a wide class of random parameterized circuits: the probability that the gradient along a given direction is nonzero, to any fixed precision, is exponentially small in the number of qubits. They tie that barren-plateau phenomenon to how close the circuit is to a 2-design. A hardware-efficient ansatz of random gates is the example they warn against, not a default that scales.

Structured ansatzes avoid that particular argument only when they actually have structure the theorem does not cover. They bring their own costs: depth, parameter count, and whether the classical optimizer can move.

Applications and limits

The application people mean is the electronic ground-state energy, or a simple spin model. The limit is everything between the variational principle and a number a chemist would use: the ansatz must contain a good state, the shot budget must resolve the energy differences that matter, noise must not wash the estimate out, and the optimizer must not stall on a barren plateau or a merely local minimum. Classical methods — exact diagonalization for tiny systems, and coupled-cluster, density-functional, and tensor-network methods beyond that — remain the tools those same molecules are usually treated with. A VQE number is interesting when it is compared with those methods on a system where the comparison is fair.

QAOA shares the hybrid loop and the ansatz language. It optimizes a combinatorial cost function rather than a molecular Hamiltonian, and it has a proved approximation ratio in one graph problem. VQE does not have an analogous guarantee for chemistry.

Reading the source

Read Peruzzo et al. for the algorithm and the HeH+ experiment. Read McClean et al. (2016) before repeating a measurement-cost claim; the formulas depend on the estimator. Read Kandala et al. for what a superconducting demonstration on LiH and BeH2 looked like in 2017. Read McClean et al. (2018) before treating a random layered ansatz as a scalable default. Aspuru-Guzik et al. (2005) is the phase-estimation chemistry proposal VQE was designed to need less coherence than.