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Theory of variational quantum simulation

Xiao Yuan, Suguru Endo, Qi Zhao, Ying Li, Simon Benjamin

arXiv:1812.08767v4quant-ph

TL;DR

Variational quantum simulation needs a unified treatment beyond pure-state unitary evolution, especially for mixed states and general stochastic dynamics. This paper compares the main variational principles, extends them to mixed-state real and imaginary time evolution, and identifies McLachlan’s principle as the most consistent framework for real-parameter quantum-gate ansätze.

  • Problem

    Existing variational quantum simulation theory largely focused on pure-state unitary evolution, while universal fault-tolerant quantum computers remained difficult to realize.

  • Method

    The paper compares three variational principles and extends variational simulation to mixed states under general real- and imaginary-time stochastic evolution with quantum-circuit realizations.

  • Results

    McLachlan’s variational principle is identified as the most consistent choice for quantum gates controlled by real parameters, including mixed-state real and imaginary time evolution.

  • Takeaways & Limitations

    The work provides a unified theory of variational quantum simulation for general real- and imaginary-time evolution applicable to near-term quantum hardware.

  • Takeaways & Limitations

    Time-dependent variational-principle evolution can become unstable because the imaginary part of its A matrix is more likely to be singular.

Abstract

from arXiv · show

The variational method is a versatile tool for classical simulation of a variety of quantum systems. Great efforts have recently been devoted to its extension to quantum computing for efficiently solving static many-body problems and simulating real and imaginary time dynamics. In this work, we first review the conventional variational principles, including the Rayleigh-Ritz method for solving static problems, and the Dirac and Frenkel variational principle, the McLachlan's variational principle, and the time-dependent variational principle, for simulating real time dynamics. We focus on the simulation of dynamics and discuss the connections of the three variational principles. Previous works mainly focus on the unitary evolution of pure states. In this work, we introduce variational quantum simulation of mixed states under general stochastic evolution. We show how the results can be reduced to the pure state case with a correction term that takes accounts of global phase alignment. For variational simulation of imaginary time evolution, we also extend it to the mixed state scenario and discuss variational Gibbs state preparation. We further elaborate on the design of ansatz that is compatible with post-selection measurement and the implementation of the generalised variational algorithms with quantum circuits. Our work completes the theory of variational quantum simulation of general real and imaginary time evolution and it is applicable to near-term quantum hardware.

1 Introduction

The introduction motivates variational quantum simulation as a near-term approach to problems beyond classical variational methods and presents a theory extending variational principles to mixed states and general stochastic evolution.

  • Motivation: Variational methods reduce exponential simulation difficulty by restricting trial states to a physically motivated subset of Hilbert space.Direct classical simulation becomes generally impossible as Hilbert-space size grows exponentially with system size.
  • Motivation: Highly entangled many-body states can defeat classical representations, motivating quantum simulation with universal quantum computers.Realising a universal fault tolerant quantum computer remains challenging.
  • Variational quantum simulation: Variational quantum simulation uses quantum processors for classically intractable cores while assigning relatively easy tasks to classical computers.This hybrid strategy targets potential quantum advantages on noisy intermediate-scale quantum hardware.
  • Scope and contribution: The work studies the equivalence and differences among Dirac–Frenkel, McLachlan, and time-dependent variational principles for quantum dynamics.These principles are established tools for classical variational simulation of dynamics.
  • Scope and contribution: The theory extends variational simulation to mixed states under general stochastic evolution and imaginary-time dynamics, including variational Gibbs state preparation.For pure-state unitary evolution, the derived correction term accounts for global phase alignment; McLachlan’s principle yields a consistent mixed-state theory.

2 Preliminary

This section introduces variational classical and quantum simulation, covering Rayleigh–Ritz ground-state estimation, parameterized quantum ansätze, and variational principles for real-time dynamics. It also explains real-parameter constraints, global-phase corrections, and simulation-error verification.

  • Rayleigh–Ritz: Rayleigh–Ritz estimates the ground-state energy by minimizing over a restricted subset of Hilbert space, avoiding exponentially hard searches of the full space.The restriction produces an approximate solution because Hilbert-space size grows exponentially with system size.
  • Quantum ansatz: In VQS, a normalized trial state is prepared with parameterized gates, and its energy is estimated by measuring Hamiltonian terms before classical parameter optimization.The ansatz state is prepared from an initial state using gates R_k(θ_k), with real parameters θ_k.
  • Real-time dynamics: Real-time variational simulation projects Schrödinger evolution onto the trial-state manifold by evolving parameters to minimize the mismatch between parameter variation and exact dynamics.The Dirac–Frenkel, McLachlan, and time-dependent variational principles provide related ways to perform this projection.
  • Variational principles: McLachlan’s principle yields real parameter evolution for real quantum-simulation parameters and is equivalent to Dirac–Frenkel evolution when complex parameters are allowed.The equivalence also holds for sufficiently expressive trial states that represent the target state.
  • Global phase: Time-dependent global-phase mismatch can invalidate parameter evolution even when trial and target states agree up to phase, but measurable correction terms resolve the problem.The same correction terms arise from applying McLachlan’s principle to mixed states.
  • Stability: Using the imaginary part of the A matrix can make the evolution matrix more likely to be singular, thereby destabilizing the variational evolution.This equation remains equivalent to the Dirac–Frenkel form when the trial state is sufficiently expressive.

3 Equivalence of the three variational principles

With complex parameters, the three variational principles yield the same parameter evolution, while real parameters can produce distinct equations. From a mixed-state perspective, the time-dependent variational principle cannot be consistently derived, supporting McLachlan’s principle as the most consistent choice for variational quantum simulation.

  • Complex parameters make the three variational principles produce the same evolution equation for parameters.
  • The time-dependent variational principle can fail when its matrices are singular or ill-conditioned, even when other principles simulate the evolution.In the single-qubit example, AI and CR are zero, making Eq. (17) unusable; generally, AI’s zero diagonal elements can make its inverse unstable.
  • For mixed states, Eq. (17) cannot be obtained consistently, whereas a variant of Eq. (12) can be derived.
  • The mixed-state derivation supports McLachlan’s principle as the most consistent variational principle for variational quantum simulation.

4 Real time evolution: open quantum systems

This section extends variational quantum simulation to mixed states under general stochastic evolutions, covering open systems and noisy hardware. It derives parameter-evolution methods, compares variational principles, and explains the global-phase correction needed for consistent pure-state unitary dynamics.

  • Mixed-state stochastic evolution: VQS is extended to mixed states under general stochastic evolution, enabling simulation of open quantum systems and noisy quantum hardware.The approach variationally simulates the stochastic master equation through parameterised mixed states.
  • Parameter evolution: Projecting stochastic evolution onto the ansatz tangent space transforms the density-matrix equation into evolution equations for the parameters.The relevant quantities M and V can be evaluated with quantum circuits.
  • Algorithm verification: The variational distance can be verified by measuring M, V, and Tr[L(ρ)^2].This provides an experimentally accessible check of the variational algorithm’s accuracy.
  • Variational principles: For complex parameters, the three variational principles are equivalent; for real parameters, McLachlan and Dirac–Frenkel remain equivalent, while time-dependent variation is trivial.Thus, McLachlan’s principle consistently produces real parameter evolution in VQS.
  • Pure-state unitary limit: Global-phase overlap terms distinguish the mixed-state-derived pure-state equations from direct McLachlan dynamics, enabling exact unitary simulation when the ansatz represents all pure qubit states up to phase.Direct application generally fails in the illustrated single-qubit example, whereas Eq. (35) is recommended for pure-state unitary real-time evolution.

5 Imaginary time evolution

Variational principles can simulate imaginary-time evolution by evolving parameters within a trial-state space. For mixed states, the resulting parameter equations remain real for real parameters, enabling variational thermal-state preparation.

  • Pure-state evolution: Imaginary-time evolution is simulated by evolving parameters of a normalised parametrised trial state.The construction applies variational principles to the unphysical imaginary-time Schrödinger evolution.
  • Pure-state evolution: For complex parameters, the three variational principles remain equivalent, whereas with real parameters only McLachlan’s principle guarantees a real evolution solution.Other formulations can produce imaginary, nonphysical parameter derivatives when parameters are restricted to real values.
  • Mixed-state evolution: Mixed-state imaginary-time evolution is obtained by projecting the density-matrix equation onto parameter variations using the Dirac–Frenkel principle.This yields a parameter equation involving the matrix M and vector Y.
  • Mixed-state evolution: For mixed states with real parameters, M and Y are real, so the parameter derivatives are real; McLachlan minimisation likewise always gives real solutions.This behavior is opposite to the pure-state case under real-parameter restrictions.
  • Variational Gibbs-state preparation: Evolving the maximally mixed state under imaginary time prepares a Gibbs state with temperature T = 1/2τ.A purification can be evolved on a combined system so that the reduced state becomes thermal.

6 Implementation

This section describes quantum-circuit implementations for pure and mixed trial states, including post-selection, and methods for measuring the coefficients required by general evolution equations. It also demonstrates excellent agreement with exact open-system dynamics, with deviation below 10^-2.

  • Pure-state preparation: Pure trial states use parameterized gate circuits, with parameterized measurements equivalently implemented by gates before fixed measurements.Post-selection uses a rank-one projector and an ancilla measurement outcome.
  • Pure-state preparation: Post-selected preparation forms a joint system-ancilla state and accepts the trial state with probability p(θ)=|⟨φ(θ)|AE|0⟩E|^2.The joint state is prepared as |φ(θ)⟩=R(θ)|0̄⟩A|0̄⟩E.
  • Mixed-state preparation: Mixed states under unitary evolution can be simulated by evolving the density matrix directly, using a purification, or separately evolving pure-state components.Separate pure-state evolution can be more accurate because each component may use different ansatz parameters, but it minimizes the distance of the whole state.
  • General evolution: Stochastic and imaginary-time evolution requires ancillae and joint parameterized circuits, because evolving pure-state components separately does not reproduce the evolution of their mixture.These evolutions are generally nonlinear on the initial state, so ρ(T)≠Σ_i p_iρ_i(T).
  • Coefficient measurement: The coefficients M, V, and Y are efficiently measured with quantum circuits, while derivative-estimation precision depends on system size and measurement count.For post-selected states, both p(θ) and ∂p(θ)/∂θ_k are directly measurable.

7 Discussion · Appendix

The paper develops the theory of variational quantum simulation, identifying McLachlan’s principle as the most consistent for real-parameter quantum gates and extending the framework to mixed-state real and imaginary time evolution. Future work includes problem-specific trial-state design, open-system and Gibbs-state simulations, experiments on near-term hardware, and error mitigation using shallow circuits.

  • 7 Discussion: The work focuses on the theory of variational quantum simulation and compares three variational principles.It studies their equivalence and differences.
  • 7 Discussion: McLachlan’s variational principle is identified as the most consistent for quantum gates controlled by real parameters.
  • 7 Discussion: The framework extends variational quantum simulation from pure states to general mixed-state evolution under real and imaginary time.
  • 7 Discussion: Future studies could design trial states for specific problems and test the theory on open quantum systems and Gibbs-state preparation.
  • 7 Discussion: Experimental realization of the variational simulation algorithms with current and near-term noisy quantum hardware remains an open direction.
  • 7 Discussion: Because the variational method uses shallow quantum circuits, error-mitigation techniques can be applied to suppress errors.

A Real time evolution … A.2 Mixed states and general evolution

The section formulates variational real-time evolution for parametrized trial states by projecting Schrödinger dynamics onto the ansatz tangent space. It also presents McLachlan’s and time-dependent variational principles through parameter-evolution equations and their associated matrix elements.

  • A.1.1 The Dirac and Frenkel variational principle: A parametrized trial state |φ(⃗θ(t))⟩ represents real-time Schrödinger evolution within the trial-state space.The parameters are collected as ⃗θ(t) = (θ1(t), θ2(t), . . . , θN(t)).
  • A.1.1 The Dirac and Frenkel variational principle: The variational construction projects the Schrödinger-equation right-hand side onto the tangent space of |φ(⃗θ(t))⟩.The projected equation is motivated by identifying the parameter derivative as a tangent vector.
  • A.1.1 The Dirac and Frenkel variational principle: Matrix elements of M and V reduce the projected dynamics to an evolution equation for the variational parameters.The passages define M and V before stating the simplified parameter evolution.
  • A.1.1 The Dirac and Frenkel variational principle: The same parameter-evolution equation follows by applying the Dirac and Frenkel variational principle.The equivalence is stated after the simplified evolution equation and is followed by a verification.
  • A.1.2 McLachlan’s variational principle: McLachlan’s variational principle minimizes the squared norm of the real-time residual (d/dt + iH)|φ(⃗θ(t))⟩.Variation of the norm is treated as equivalent to variation of its square.
  • A.1.2 McLachlan’s variational principle: McLachlan’s principle yields a corresponding evolution equation for the variational parameters.The parameter equation is stated after defining the residual objective.
  • A.1.2 McLachlan’s variational principle: The relevant parameter equations use the real and imaginary parts of the matrix elements Ai,j and Ci.The passages specify these real and imaginary components for both McLachlan’s and the time-dependent formulations.
  • A.1.3 Time-dependent variational principles: The time-dependent variational formulation derives the Schrödinger equation from a Lagrangian and gives an equivalent parameter-evolution equation.The formulation separately identifies the Lagrangian, its variational derivation, and the resulting parameter dynamics.

A.2.1 General evolution

This section develops variational parameter evolution for mixed-state stochastic dynamics using tangent-space projection, McLachlan minimization, and the time-dependent variational principle. It also notes that restricting parameters to be real can prevent reproducing the stochastic evolution.

  • Parameterised trial states: A parameterised trial state ρ(⃗θ(t)) provides parameters whose evolution simulates the stochastic evolution.The construction begins from a mixed-state evolution and seeks the corresponding evolution of θ.
  • Time-dependent variational principle: The Dirac–Frenkel principle projects the stochastic evolution equation onto the trial-state tangent subspace.The projection is formulated using derivatives with respect to the variational parameters.
  • McLachlan’s variational principle: McLachlan’s variational principle minimizes the distance between parameterized and stochastic evolutions and yields the same parameter equation.The distance can equivalently be minimized using its squared norm.
  • Real-parameter limitation: When parameters θ are real, the stochastic evolution equation cannot be reproduced, so the parameter evolution equation cannot be obtained.By contrast, for real θ, the matrices M and V and the solution θ̇j are also real when the evolution equation is available.

A.2.2 Reduction to the pure state case … B.2.2 Reduction to the pure state case

The paper extends variational simulation to imaginary-time mixed-state evolution and shows that both real- and imaginary-time mixed-state equations reduce to pure-state evolution with an ancilla phase correction. It relates Dirac–Frenkel, McLachlan, and time-dependent variational formulations while identifying restrictions for real parameters.

  • A.2.2 Reduction to the pure state case: Unitary mixed-state evolution is governed by the von Neumann equation L(ρ) = −i[H, ρ(t)].
  • A.2.2 Reduction to the pure state case: For pure states, the von Neumann equation becomes the Schrödinger equation, and the variational evolution follows the corresponding pure-state formulation.
  • A.2.2 Reduction to the pure state case: An additional ancilla phase gate with parameter θ0 makes the pure-state evolution equivalent to the mixed-state evolution.The equivalence adds one ancilla parameter to an ansatz with N parameters.
  • B.1.1 The Dirac and Frenkel variational principle: Imaginary-time evolution is obtained by Wick rotation τ = it, with normalized parametrized trial states used to derive parameter evolution.
  • B.1.1 The Dirac and Frenkel variational principle: The Dirac–Frenkel principle produces the same simplified parameter-evolution equation for imaginary-time pure-state simulation.
  • B.1.2 McLachlan’s variational principle: McLachlan’s principle derives an imaginary-time parameter equation by minimizing the evolution residual, using the real parts of matrix elements when parameters are complex.
  • B.1.3 Time-dependent variational principles: The time-dependent variational principle modifies the Lagrangian using normalization and Eτ = ⟨ψ(τ)|H|ψ(τ)⟩, but real parameters yield an incorrect imaginary solution for ˙θj.
  • B.2.1 Evolution of mixed states: For mixed states, imaginary-time evolution follows the anticommutator equation, and the Dirac–Frenkel, McLachlan, and time-dependent principles produce corresponding parameter equations.When parameters are real, M and Y are real and the solution ˙⃗θ is real; real parameters cannot reproduce the imaginary-time evolution because ρ = ρ†.
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