Variational Quantum Circuit (VQC)

• Variational quantum circuits (VQCs) are parameterized quantum circuits optimized with classical algorithms to prepare complex quantum states efficiently.
• The telescoping Hamiltonian and history state constructions transform circuit simulation into minimization problems, ensuring scalable and error-resilient implementations.
• Tailored for NISQ devices, VQCs leverage shallow circuits and measurement-based optimizations, balancing resource efficiency with universal quantum computation.

A variational quantum circuit (VQC) is a parameterized quantum circuit designed as a quantum subroutine within a hybrid quantum–classical computational framework, in which circuit parameters are optimized using classical algorithms to minimize (or maximize) objective functions evaluated via quantum measurement. VQCs are central to quantum-enhanced algorithms for optimization, state preparation, simulation, and machine learning, particularly in the noisy intermediate-scale quantum (NISQ) era where circuit depth and quantum coherence are limited.

1. Universal Variational Quantum Computation and Objective Functions

VQCs gain their computational power by embedding the search for complex quantum states or the implementation of quantum algorithms into the minimization of an objective function. The theoretical foundation for the universality of variational quantum computation is established by constructing two distinct classes of objective functions whose minimization prepares the output of any desired quantum circuit (Biamonte, 2019):

1. Telescoping Hamiltonian Construction: This approach starts from a simple projector penalizing deviations from the initial product state, typically the all-zeros state $|0\rangle^{\otimes n}$. The telescoped Hamiltonian is constructed iteratively as

$$h(k) = \left(\prod_{\ell=1}^k U_\ell\right) P_\phi \left(\prod_{\ell=1}^k U_\ell\right)^\dagger,\;\;\;(k = 1, \ldots, L)$$

where $U_\ell$ are the target circuit’s unitary gates, $P_\phi$ is a projector (e.g., $$P_\phi = \sum_{i=1}^n |1\rangle\langle1|^{(i)} = \frac{n}{2}[1 - \frac{1}{n}\sum_{i=1}^n Z^{(i)}]$$), and $L$ is the total circuit depth. The target state emerges as the unique ground state of $h(L)$. The general form of the objective function decomposes into the Pauli basis:

$$\mathcal{H} = \sum \mathcal{J}^{a_1\ldots a_n}_{\alpha_1\ldots\alpha_n} \sigma^{(a_1)}_{\alpha_1} \ldots \sigma^{(a_n)}_{\alpha_n}\,.$$

Achieving an expectation value below a chosen threshold $\Delta$ certifies state preparation fidelity.

1. History State (Feynman–Kitaev Clock) Construction: The circuit is "unrolled" over a time register, introducing $O(\log L)$ slack qubits, and encoding the computation’s history as

$$|\psi_\mathrm{hist}\rangle = \frac{1}{\sqrt{L+1}} \sum_{t=0}^L \left(\prod_{\ell=1}^t U_\ell\right) |\xi\rangle \otimes |t\rangle\,,$$

where $|\xi\rangle$ is an input register and $|t\rangle$ denotes the clock. The propagation Hamiltonian

$$\mathcal{H}_\mathrm{prop} = \sum_{t=1}^L \left[-U_t \otimes |t\rangle\langle t{-}1| - U^\dagger_t \otimes |t{-}1\rangle\langle t| + |t\rangle\langle t| + |t{-}1\rangle\langle t{-}1|\right]$$

together with an input penalty $\mathcal{H}_\mathrm{in}$, forms a total Hamiltonian $$\mathcal{H}_\mathrm{total} = J\mathcal{H}_\mathrm{in} + K\mathcal{H}_\mathrm{prop}$$ with a unique ground state encoding the computation. Measurement of the clock register in state $|L\rangle$ yields the circuit output.

2. Classical-Quantum Hybrid Optimization Loop

The VQC paradigm employs a hybrid iterative loop involving quantum execution and classical parameter optimization (Biamonte, 2019):

• The parameterized VQC (ansatz) is executed to prepare a state $|\psi(\theta)\rangle$ for current parameter set $\theta$.
• The cost/objective function $$\langle\psi(\theta)|\mathcal{H}|\psi(\theta)\rangle$$ is estimated by quantum measurement, typically via local Pauli basis measurements decomposed term-wise.
• Classical optimization updates $\theta$ to minimize the objective value, using measured statistics.
• Iteration continues until the objective value crosses a predefined tolerance $\Delta$, at which point, due to the variational principle and associated stability lemmas, the prepared quantum state exhibits high overlap with the desired target state.

This outer loop approach greatly reduces the need for coherent quantum circuit depth, with the "fleeting resource" being the number of quantum observable evaluations required.

3. Resource Efficiency and Scaling

The efficiency and scalability of the VQC model stem from the structure of the chosen objective function and properties of the target circuit:

• Telescoping Construction Efficiency: Clifford gates conjugate Pauli strings to Pauli strings without increasing the number or algebraic degree of Pauli terms. Thus, the number of expected value evaluations needed to achieve convergence is insensitive to the Clifford portion of the circuit; only non-Clifford gates can increase the term count, typically by at most $4^2$ per gate acting on two qubits. The number of non-Clifford gates must therefore remain $O(\mathrm{poly}(\log n))$ for efficient scaling, guaranteeing the total number of measurements is $O(\mathrm{poly}(n,L))$.
• History State Construction Efficiency: Requires $O(L^2)$ expected value estimations carried on $n + O(\log L)$ qubits, where $L$ is the number of circuit gates. The polynomial measurement overhead is independent of (or only weakly dependent on) the depth of most Clifford gates due to efficient clock register encoding and term grouping in the propagation Hamiltonian.

This polynomial-in-$n$ or $L$ scaling ensures practical feasibility for quantum circuits beyond the capacity of brute-force, deep quantum implementations.

4. Universality and Model-Theoretic Implications

By constructing objective functions whose minimization is equivalent to the execution of any desired quantum circuit, the variational paradigm is shown to be computationally universal (Biamonte, 2019):

• For any target unitary circuit, there exists a VQC with an associated objective such that minimization to below a threshold $\Delta$ ensures high-fidelity preparation of the correct output state.
• Both the telescoping and history state constructions admit hybrid optimization, removing the necessity for deep, coherent gate sequences. Thus, any quantum computation can, in principle, be simulated (to arbitrary accuracy) within the VQC framework using polynomially many observable evaluations and local measurements.
• This result elevates variational quantum computation to the same formal standing as traditional circuit models, fundamentally broadening the landscape of quantum programming.

The implication is an expansion of accessible quantum algorithms—directly leveraging state-preparation and measurement—that are naturally compatible with NISQ hardware constraints.

5. Practical Implementation and NISQ Relevance

The VQC framework is tailored for near-term quantum devices with limited coherence and gate fidelity:

• Instead of executing deep gate sequences, variational quantum programming enables the preparation of complex quantum states, many-body ground states, or algorithmic outputs through shallow, parameterized circuits and adaptive, measurement-based optimization loops.
• The local measurement structure and term-wise cost function evaluation are resilient to errors and do not require long coherent sequences, making them well-suited to NISQ platforms.
• By ensuring that most operations are Clifford (and thus practically "free" in terms of measurement overhead), only a small overhead for non-Clifford operations is incurred; this supports the execution of meaningful quantum programs within noisy environments.

6. Synthesis and Broader Context

The formalism developed in (Biamonte, 2019) integrates the variational principle with Hamiltonian constructions that "compile" quantum circuits into objective-driven state preparation and optimization. By showing that appropriately constructed variational objectives can simulate arbitrary quantum circuits, variational quantum computation is proven to be a universal model, competitive with traditional circuit-based approaches but more amenable to the constraints and opportunities of contemporary quantum hardware.

This result positions the VQC paradigm as a foundation for both current and future quantum algorithm design, emphasizing hybrid classical-quantum workflows, flexible measurement protocols, and scalable, hardware-adaptable circuit ansatze. The operational focus shifts from deep gate sequences to efficient optimization of expectation values, aligning with both the physics of NISQ devices and the mathematical structure of quantum computational universality.