Variational Quantum Algorithms

• Variational Quantum Algorithms are hybrid methods using parameterized quantum circuits and classical processors to simulate complex quantum dynamics.
• They apply McLachlan’s variational principle to project evolution equations onto a tractable parameter space, making them suitable for NISQ devices.
• Practical applications include ground-state preparation, optimization tasks, and open-system simulations, validated by benchmarks on dissipative Ising models.

Variational quantum algorithms (VQAs) are a class of hybrid quantum–classical algorithms in which a quantum computer prepares parameterized quantum states while a classical processor iteratively refines these parameters to optimize a problem-specific objective function. VQAs are designed to leverage shallow quantum circuits amenable to noisy intermediate-scale quantum (NISQ) devices and are broadly applicable to tasks including ground-state preparation, dynamical simulation, optimization, and even simulation of open quantum systems. Central to this methodology is the use of a parameterized quantum circuit (ansatz) and an adaptive variational principle (notably, McLachlan’s) to project quantum evolution or algebraic problems into tractable updates in the parameter space. The framework is sufficiently general to encompass closed dynamics, algebraic matrix problems, and Markovian open-system evolution.

1. Unified Variational Framework for Generalized Evolution

A general, operator-theoretic perspective recasts quantum (and certain non-Hermitian or dissipative) evolution as a first-order, operator-valued differential equation:

$$B(t) \frac{d}{dt} |v(t)\rangle = \sum_{j} A_j(t) |v'_j(t)\rangle$$

where $B(t)$ and $A_j(t)$ are typically sparse, possibly non-Hermitian operators, and the $|v'_j(t)\rangle$ may differ from $|v(t)\rangle$. This subsumes standard real- and imaginary-time Schrödinger evolution, Lindblad-type open system evolution, and linear matrix problems.

To implement this variationally, one assumes

$$|v(\vec{\theta}(t))\rangle = \alpha(\theta_0(t))\, |\phi(\vec{\theta}_1(t))\rangle$$

where $|\phi(\vec{\theta}_1)\rangle$ is a parameterized ansatz circuit state and $\vec{\theta}$ are the real-valued parameters. Using McLachlan’s variational principle, the algorithm projects the evolution onto the tangent space of ansatz states, yielding coupled equations of motion for $\vec{\theta}$:

$$\sum_j \widetilde{B}_{k,j}\, \frac{d\theta_j}{dt} = \widetilde{V}_k$$

with

$$\widetilde{B}_{k,j} = \operatorname{Re}\left\{ \frac{\partial \langle \phi(\vec{\theta}) |}{\partial \theta_k} B^\dagger B \frac{\partial |\phi(\vec{\theta})\rangle}{\partial \theta_j} \right\}$$

and analogous explicit forms for $\widetilde{V}_k$ incorporating $A_j(t)$, $B(t)$, and derivatives of the ansatz. For Hermitian $B(t)$ and particular $A_j$, this reduces to the equations of standard real- and imaginary-time variational simulation.

2. Linear Algebra Problems via Variational Dynamics

Linear algebraic problems such as solving $M|v_0\rangle = |v_m\rangle$ and generic matrix–vector multiplications are mapped into a dynamical interpolation task. Specifically, the evolution

$$|v(t)\rangle = E(t)|v_0\rangle,\quad E(t) = (t/T)\, M + [1 - (t/T)]\, I$$

with $|v(0)\rangle = |v_0\rangle$, $|v(T)\rangle = M|v_0\rangle$, is governed by

$$\frac{d}{dt} |v(t)\rangle = G |v_0\rangle,\quad G = (M - I)/T$$

and fits into the general Eq. (1) with $A(t) = G,\, B(t) = I$. This dynamical reformulation allows variational simulation of otherwise static matrix–vector computations.

If $M$ has a tensor product form—$M = M_1 \otimes \ldots \otimes M_\ell$—one applies singular value decomposition $M = U D V$, where $U$ and $V$ are unitaries and $D$ is diagonal. Unitary parts are realized as exponentials (e.g. $U = \exp( -i H^U T^U )$); thus real–time variational evolution is used. The diagonal, possibly non-unitary $D$ is simulated by “unnormalized” imaginary–time evolution, $$\partial_\tau |v(\tau)\rangle = -H^D |v(\tau)\rangle$$, where $D \approx \exp( -H^D T^D )$. In this way, real and imaginary time evolutions are variationally combined to efficiently realize matrix multiplication operations that otherwise would not be scalable for generic $M$.

3. Variational Tensor Strategies: Hybrid Real and Imaginary Time

For tensor-product Hamiltonians and similar problems, further resource reduction is gained by treating the unitary and non-unitary components via distinct evolution strategies:

• Unitary elements $U, V$ are generated as time evolutions: for any unitary $$U = \sum_j e^{i \lambda_j} |\lambda_j\rangle \langle \lambda_j|$$, this can be written

$$U = \exp( -i H^U T^U ),\quad H^U = - \sum_j \frac{\lambda_j}{T^U} |\lambda_j\rangle \langle \lambda_j|$$

Variational real–time evolution simulates this process within the ansatz manifold.

• Diagonal matrices $D$ are embedded into imaginary–time evolution:

$D \approx \exp(- H^D T^D )$

with $H^D$ defined by logarithmic spectral content of $D$. This normalization is controlled to prevent divergence for zero eigenvalues. The variational simulation transitions from the initial to target vector as in the linear algebra mapping.

The hybrid use of real and imaginary time in variational circuits enables more general classes of operations, beyond those accessible by real– or imaginary–time VQS alone.

4. Simulation of Open Quantum System Dynamics

Open system evolution, formulated via the Lindblad master equation

$$\frac{d\rho}{dt} = -i[H,\rho] + \mathcal{L}\rho, \quad \mathcal{L}\rho = \sum_k \frac{1}{2} (2 L_k \rho L_k^\dagger - L_k^\dagger L_k \rho - \rho L_k^\dagger L_k)$$

is addressed via a stochastic Schrödinger equation for quantum trajectories:

$$d|\psi_c(t)\rangle = \left[ -iH - \frac{1}{2} \sum_k (L_k^\dagger L_k - \langle L_k^\dagger L_k\rangle) \right] |\psi_c(t)\rangle\, dt + \sum_k \left( \frac{L_k |\psi_c(t)\rangle}{\|L_k |\psi_c(t)\rangle\|} - |\psi_c(t)\rangle \right) dN_k$$

Each trajectory alternates between continuous, non-Hermitian variational evolution (using the generalized framework described above) and stochastic “jumps” corresponding to discrete quantum events. Variational matrix–vector multiplication is used at every jump step for state update via $L_k$, using the previously described tensor structure approaches.

Algorithmically, pseudocode in the original work details time-step procedures: random numbers are drawn at each step, and the state undergoes either a jump or smooth evolution according to the jump probabilities calculated from expectation values. This permits simulation of open-system dynamics with NISQ-suitable, shallow circuit ansätze without explicit density-matrix manipulation or purification.

5. Numerical Validation and Performance Assessment

Demonstrative simulations target a six-qubit, two-dimensional transverse-field Ising model under dissipation. The Hamiltonian is

$$H_I = \frac{J}{4} \sum_{\langle ij \rangle} Z_i Z_j + h_X \sum_{i=1}^6 X_i$$

with Lindblad operators $L_i = \sqrt{\gamma} \sigma^+_i$ and $\sigma^+_i = |1\rangle\langle 0|_i$, where $J = h_X = \gamma = 1$; the initial state is $|0\rangle^{\otimes 6}$. Evolution is simulated for $t \in [0,6]$, monitoring the observable

$$C = \frac{1}{7} \sum_{\langle ij \rangle} \langle Z_i Z_j \rangle$$

The variational ansatz is a symmetry-adapted Hamiltonian ansatz sandwiched by layers of X rotations, for a total of 54 parameters. The variational simulations reproduce the exact solution (unitary and dissipative) with maximal errors around $0.01$. Signatures of dissipation-induced dynamical transitions—such as the decay of nearest-neighbor correlations—match both theory and experiment, showing the ansatz is sufficiently expressive and trainable for NISQ applications.

6. Summary and Broader Significance

The variational quantum simulation framework discussed here provides several unifying and operational advances:

• It recasts a diverse set of processes (real/imaginary/non-Hermitian evolution, algebraic linear operations, open-system stochasticity) into a single variational evolution structure by generalizing the parameter space projection via McLachlan’s principle.
• It enables static problems (e.g., linear system solution, matrix–vector multiplication) to be performed as variational dynamical simulations, with further efficiency for tensor-structured matrices.
• For open-system Lindblad evolution, it implements a quantum-jump unraveling that combines variational evolution and variationally realized quantum jumps in a shallow-circuit architecture.
• In practical benchmarks, it achieves $\sim$0.01 maximal error on six-qubit, two-dimensional dissipative Ising dynamics with symmetry-preserving, shallow ansätze tailored for NISQ circuits.

This approach thus establishes variational quantum algorithms as a unified, resource-efficient, and scalable tool for a wide range of quantum simulation and algebraic tasks—closed and open—on near-term devices (Endo et al., 2018).