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Variational quantum algorithms


Applications such as simulating complicated quantum systems or solving large-scale linear algebra problems are very challenging for classical computers, owing to the extremely high computational cost. Quantum computers promise a solution, although fault-tolerant quantum computers will probably not be available in the near future. Current quantum devices have serious constraints, including limited numbers of qubits and noise processes that limit circuit depth. Variational quantum algorithms (VQAs), which use a classical optimizer to train a parameterized quantum circuit, have emerged as a leading strategy to address these constraints. VQAs have now been proposed for essentially all applications that researchers have envisaged for quantum computers, and they appear to be the best hope for obtaining quantum advantage. Nevertheless, challenges remain, including the trainability, accuracy and efficiency of VQAs. Here we overview the field of VQAs, discuss strategies to overcome their challenges and highlight the exciting prospects for using them to obtain quantum advantage.

Key points

  • Variational quantum algorithms (VQAs) are the leading proposal for achieving quantum advantage using near-term quantum computers.

  • VQAs have been developed for a wide range of applications, including finding ground states of molecules, simulating dynamics of quantum systems and solving linear systems of equations.

  • VQAs share a common structure, where a task is encoded into a parameterized cost function that is evaluated using a quantum computer, and a classical optimizer trains the parameters in the VQA.

  • The adaptive nature of VQAs is well suited to handle the constraints of near-term quantum computers.

  • Trainability, accuracy and efficiency are three challenges that arise when applying VQAs to large-scale applications, and strategies are currently being developed to address these challenges.

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Fig. 1: Applications of variational quantum algorithms.
Fig. 2: Schematic diagram of a variational quantum algorithm.
Fig. 3: Schematic diagram of an ansatz.
Fig. 4: Variational quantum eigensolver implementation.
Fig. 5: Quantum approximate optimization algorithm.
Fig. 6: Barren plateau phenomenon.


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M.C. thanks K. Sharma for discussions. M.C. was initially supported by the Laboratory Directed Research and Development (LDRD) programme of Los Alamos National Laboratory (LANL) under project no. 20180628ECR, and later supported by the Center for Nonlinear Studies at LANL. A.A. was initially supported by the LDRD programme of LANL under project no. 20200056DR, and later supported by the US Department of Energy (DOE), Office of Science, Office of High Energy Physics QuantISED programme under contract nos. DE-AC52-06NA25396 and KA2401032. S.C.B. acknowledges financial support from EPSRC Hub grants under the agreement nos. EP/M013243/1 and EP/T001062/1, and from EU H2020-FETFLAG-03-2018 under the grant agreement no. 820495 (AQTION). S.E. was supported by MEXT Quantum Leap Flagship Program (MEXT QLEAP) grant nos. JPMXS0120319794, JPMXS0118068682 and JST ERATO grant no. JPMJER1601. K.F. was supported by Japan Society for the Promotion of Science (JSPS) KAKENHI grant no. 16H02211, JST ERATO JPMJER1601 and JST CREST JPMJCR1673. K.M. was supported by JST PRESTO grant no. JPMJPR2019 and JSPS KAKENHI grant no. 20K22330. K.M. and K.F. were also supported by MEXT QLEAP grant no. JPMXS0118067394 and JPMXS0120319794. X.Y. acknowledges support from the Simons Foundation. L.C. was initially supported by the LDRD programme of LANL under project no. 20190065DR, and later supported by the US DOE, Office of Science, Office of Advanced Scientific Computing Research under the Quantum Computing Application Teams (QCAT) programme. P.J.C. was initially supported by the LANL ASC Beyond Moore’s Law project, and later supported by the US DOE, Office of Science, Office of Advanced Scientific Computing Research, under the Accelerated Research in Quantum Computing (ARQC) programme. Most recently, M.C., L.C. and P.J.C. were supported by the Quantum Science Center (QSC), a National Quantum Information Science Research Center of the US DOE.

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Supplementary information


Circuit depth

An integer number that counts the maximum length in the circuit between the input and the output. This length is usually defined in terms of layers of gates acting in parallel.

Ancilla qubits

Auxiliary qubits used during a quantum computation.

Trotterized adiabatic transformation

A quantum adiabatic transformation involves changing a system’s Hamiltonian slowly enough that the system remains in the ground state. Trotterization approximates this evolution with a series of discrete steps.

Pauli strings

A Pauli string acting on n qubits is defined as a Hermitian operator from the set {1,X,Y,Z}n, where X,Y,Z are Pauli operators, and 1 is the identity on a single qubit.


A mathematical method used to approximate the matrix etH, which describes the evolution for a time t under a Hamiltonian H as \({e}^{tH}\sim {({e}^{\frac{tH}{n}})}^{n}\). The Trotter error is the error induced from the Trotter method approximation.

Quantum optimal control

Quantum optimal control is a theoretical framework that provides tools for the systematic manipulation of quantum dynamical systems.

Rayleigh–Ritz variational principle

Principle stating that the lowest eigenvalue of a Hermitian operator H is upper-bounded by the minimum expectation value 〈ψHψ〉 found by varying the state \(| \psi \rangle \).

Hadamard test

Method used to estimate the real and complex part of the expectation value of a unitary operator over a quantum state.


The Max-Cut problem aims to find the maximum cut of a graph, that is, the partitioning of a graph’s vertices that cuts through the most edges.

Schur concavity

A Schur concave function f is a funtion \(f:{{\mathbb{R}}}^{d}\to {\mathbb{R}}\) that for all x and y in \({{\mathbb{R}}}^{d}\) is such that if x is majorized by y, then f(x) ≥ f(y).

Unitary 2-design

Ensemble of unitaries, such that sampling over their distribution yields the same properties as sampling random unitaries from the Haar distribution measure up to the first two moments.

Born machines

Generative models that represent the probability distribution of classical dataset as quantum pure states.

Restricted Boltzmann machine

Generative computational model that can learn a probability distribution over its set of inputs.

Quantum Fisher information

The quantum Fisher information quantifies the sensitivity of a quantum state to a parameter or a set of parameters.

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Cerezo, M., Arrasmith, A., Babbush, R. et al. Variational quantum algorithms. Nat Rev Phys 3, 625–644 (2021).

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