Variational Quantum Computing Approach

• Variational quantum computing is a hybrid quantum–classical method that approximates quantum states and dynamics using parameterized circuits and classical optimization.
• It leverages foundational variational principles, such as the Rayleigh–Ritz and Dirac–Frenkel methods, to derive evolution equations that respect hardware constraints.
• Optimized ansatz design and efficient measurement strategies enable practical simulation of open, finite-temperature, and many-body quantum systems on NISQ hardware.

A variational quantum computing approach leverages the variational principle to approximate quantum states and simulate quantum dynamics using parameterized quantum circuits (ansätze), coupled to classical optimization routines. This hybrid quantum–classical paradigm addresses both static and dynamical quantum many-body problems, extending to mixed states, real- and imaginary-time evolution, and is directly tailored for the capabilities of near-term noisy intermediate-scale quantum (NISQ) hardware (Yuan et al., 2018). The variational method reformulates quantum simulation as the evolution of classical parameters defining trial quantum states, yielding shallow circuit constructions amenable to hardware constraints.

1. Foundational Variational Principles in Quantum Simulation

A variational quantum simulation (VQS) framework is grounded in mapping the target quantum evolution onto a set of equations for the variational parameters characterizing a trial quantum state (pure or mixed). The standard approaches reviewed and rigorously developed include:

• Rayleigh–Ritz Principle (Static Problems): Used for ground-state energy estimation, where the variational quantum eigensolver (VQE) becomes the general quantum version.
• Dirac–Frenkel Principle (Dynamic Problems): Projects the difference between the true time derivative (from Schrödinger or quantum master equation) and the trial state derivative onto the tangent space spanned by the parameter derivatives.
• McLachlan’s Principle: Minimizes the norm $\|\dot{\psi} - (-i H \psi)\|$, yielding evolution equations optimized for real-valued parameterizations, which are typical in quantum circuits. For real parameters, this approach guarantees well-conditioned, real-valued evolution equations and is preferred for VQS when the TDVP may yield unstable behavior.
• Time-Dependent Variational Principle (TDVP): Less robust for real-valued parameters due to its sensitivity to parameter domain and potentially pathological stationary points (Yuan et al., 2018).

These principles provide the mathematical infrastructure to derive dynamical equations for the variational parameters, ensuring that the state trajectory remains as close as possible (according to the chosen metric) to the true quantum evolution.

2. Mapping Dynamics to Parameter Evolution Equations

Real-time and imaginary-time quantum dynamics are encoded variationally by translating the evolution equations of the system’s density matrix or wavefunction into differential or algebraic equations for classical parameters:

• General Form:

$\mathbf{A} \cdot \dot{\vec{\theta}} = \mathbf{F}$

where the matrix $\mathbf{A}$ and vector $\mathbf{F}$ contain quantum expectation values and overlap terms between derivatives of the trial state (or its purification) and Hamiltonian terms.

• For Mixed States with Stochastic/Nonunitary Evolution:

The VQS formalism is generalized to parameterized density matrices (or purifications) and arbitrary Lindbladian/stochastic evolutions. The evolution equations involve objects such as $M$ and $V$ matrices, corresponding to the projected increments along tangent vectors of the parameter manifold.

• Imaginary-Time VQS and Gibbs State Preparation:

Imaginary-time propagation filters out excited states, driving the system toward its ground state or—more generally—Gibbs (thermal) equilibrium. The variation is performed analogously to real time but with $\tau = it$ substitution, leading to effective cooling dynamics and non-unitarity. Parameter update equations are derived based on projection onto the tangent space consistent with the variational principle.

• Global Phase Correction:

For pure states, an alignment term is incorporated to maintain consistent global phase tracking, ensuring reduction from mixed to pure dynamics remains well defined.

This parameter evolution reduces the exponentially complex quantum simulation to a system of manageable equations for a small number of parameters, with resource requirements scaling with parameter count and measurement cost rather than Hilbert space dimension.

3. Ansatz Design: Hardware Realizability and Post-Selection

The ansatz—i.e., the specific parameterized quantum circuit—crucially determines expressivity, resource demands, and suitability for hardware:

• Gate-Based Ansatz:

Typically constructed as a composition of one- and two-qubit parameterized gates of the form $\exp(i \theta_j \sigma_j)$ (with Hermitian $\sigma_j$) acting on an initial reference state. This ensures immediate state normalization and real-valued parameters.

• Measurement and Post-Selection Compatible Ansatz:

One can extend the ansatz by generating a joint state on system and ancilla qubits, then apply measurement and rank-one projective post-selection to the ancillas. The remaining subsystem is left in the desired trial state. This approach increases the flexibility and allows realizing states otherwise inaccessible by simple gate constructions.

• Purification Techniques for Mixed States:

Mixed states, required for simulating open system dynamics or finite-temperature states, utilize purifications and extend the ansatz to joint system–environment (ancilla) circuits.

The deliberate choice and design of the ansatz directly impact optimization landscapes, experimental feasibility, and the capability to encode relevant physical symmetries or correlations.

4. Quantum Hardware Implementation and Measurement Strategies

Implementing variational quantum simulations on real quantum hardware involves efficient measurement schemes to extract the coefficients appearing in evolution equations:

• Overlap and Derivative Estimation:

Measurement of terms such as $$\langle \partial_{\theta_i} \psi | H | \psi \rangle$$ and various state overlaps appears at the core of assembling matrices $\mathbf{A}$ and vectors $\mathbf{F}$ (or their imaginary-time equivalents). Circuits for these measurements often rely on ancilla tricks (Hadamard test, swap test, or variants).

• Shallow Circuit Depth:

Because the evolution proceeds by variationally updating parameters rather than Trotterizing long-time dynamics, individual steps require only shallow circuits, compatible with NISQ-era coherence budgets.

• Measurement Overheads and Error Mitigation:

The work addresses required sampling budgets by analyzing signal-to-noise properties for matrix and vector elements. Variational VQS naturally accommodates error mitigation approaches, such as symmetry verification and extrapolation, within the classical optimization loop.

• Post-Selection Overheads:

When the ansatz involves post-selection, the measurement cost is increased, but post-selection enables access to nontrivial state spaces or enforces system constraints.

These practices collectively ensure the operation of VQS algorithms on experimentally constrained, noisy devices.

5. Unification of Real and Imaginary Time, Mixed States, and Open Quantum Dynamics

The theoretical formulation developed is unified and general, encompassing:

• Real-Time Unitary Evolution:

Formally equivalent to Schrödinger evolution, with the variational trajectory bound by the chosen principle (McLachlan, Dirac–Frenkel).

• Imaginary-Time Nonunitary Evolution:

Utilizes variational projections compatible with nonunitary propagation. The technique is essential for ground-state filtration and for time evolution into finite-temperature (Gibbs) states.

• Open System Dynamics and Stochastic Evolution:

By parameterizing the density matrix (or purification) and extending projection methods, the formalism covers arbitrary stochastic open-system quantum dynamics, essential for quantum simulation of real materials and noisy quantum devices.

• Gibbs State Preparation:

Efficient preparation of thermal states is enabled by applying imaginary-time evolution to a maximally mixed initial state or via purification on an extended Hilbert space.

This comprehensive theoretical scope provides a foundation for developing practical algorithms for quantum chemistry, condensed matter systems, materials at finite temperatures, and open quantum systems.

6. Practical Applicability and Quantum Advantage in the NISQ Era

The approach rendered in this formalism is particularly relevant for near-term devices due to:

• Circuit Shallowing:

Avoidance of long gate sequences characteristic of Trotterized simulations allows operation within error and coherence time budgets.

• Measurement-Resource Trade-Offs:

Classical parameter optimization cycles are naturally adjusted to maximize progress per quantum sample, incorporating knowledge of error rates and measurement noise.

• Integration with Error Mitigation:

Variational hybrid cycles permit flexible mitigation of noise, such as symmetry-based post-selection, inclusion of correction circuits, or incorporation of extrapolation methods to suppress bias from decoherence and measurement.

• Flexible Ansatz and Adaptive Algorithms:

The generality of the approach supports various ansatz constructions, including hardware-efficient, problem-adaptive, and symmetry-enforcing circuits, all tailored to the physical problem and available experimental resources.

These features collectively make the variational quantum computing approach a core enabling platform for quantum simulation and computation in the pre-fault-tolerant era, broadly impacting quantum chemistry, quantum materials, and dynamical quantum information science.

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In summary, the variational quantum computing approach, formalized via the framework of time-dependent and time-independent variational principles, provides a rigorous foundation for simulating both unitary and nonunitary quantum dynamics—covering pure and mixed states—using shallow, parameterized quantum circuits optimized in tandem with classical algorithms (Yuan et al., 2018). By extending the conventional variational paradigm to include open systems, mixed state dynamics, and experimentally relevant measurement/post-selection protocols, this approach offers a scalable and practical solution for key problems in quantum simulation on near-term hardware.