Variational Quantum Algorithm

• Variational quantum algorithms are parameterized quantum circuits optimized via classical feedback to minimize cost functions based on observable expectation values.
• They utilize constructions such as Telescoping and Feynman–Kitaev to map quantum circuit outputs to Hamiltonian ground states, ensuring universality of computation.
• Key challenges include managing measurement overhead and the scaling effects of non-Clifford gates to enable efficient computation on near-term quantum devices.

A variational quantum algorithm (VQA) is a quantum computational procedure employing a parameterized quantum circuit whose parameters are optimized using a classical outer-loop algorithm, typically with the objective of minimizing or maximizing a cost function related to quantum measurements. VQAs have emerged as a leading paradigm for quantum-enhanced optimization, eigenvalue estimation, quantum simulation, and, more broadly, as a model for universal quantum computation in the noisy intermediate-scale quantum (NISQ) era (Biamonte, 2019). By iteratively measuring expectation values and feeding back to a classical optimizer, VQAs can prepare outputs of arbitrary quantum circuits in a resource-efficient manner, with key cost drivers including the number of expected values to be measured, circuit depth, and optimizer performance.

1. Theoretical Principles and Universal Construction

Variational quantum computation is based on preparing a trial quantum state |ψ(θ)⟩ using an ansatz parameterized by a vector θ. The algorithm proceeds by defining and minimizing a cost function, often taking the form of an expectation value over a Hamiltonian or observable:

$$E(\theta) = \langle \psi(\theta) | \mathcal{H} | \psi(\theta) \rangle.$$

The central theoretical result is that, given suitable objective functions, VQAs are universal for quantum computation. Specifically, it is possible to engineer objective functions with a unique global minimum at the state produced by any given quantum circuit (Biamonte, 2019). Two key constructions are identified:

• Telescoping (Penalty Hamiltonian) Construction: Start with a simple penalty Hamiltonian, such as

$$P_{\phi} = \sum_{i=1}^{n} |1\rangle \langle 1|^{(i)} = \frac{n}{2}\left[1 - \frac{1}{n} \sum_{i=1}^{n} Z^{(i)} \right]$$

where $Z^{(i)}$ is the Pauli matrix acting on qubit $i$. By conjugating this penalty with the circuit gates $U_{\ell}$, yielding $$h(k) = (\prod_{\ell=1}^k U_\ell) P_\phi (\prod_{\ell=1}^k U_\ell)^\dagger$$, and extending over all gates in the target circuit, the ground state of the constructed Hamiltonian is precisely the output of the original circuit.

• Feynman–Kitaev (History State) Construction: Extend the objective via introduction of a clock register to track gate application, leading to a Hamiltonian

$$\mathcal{H} = J\mathcal{H}_{in} + K\mathcal{H}_{prop},$$

where $\mathcal{H}_{in}$ fixes the input and $\mathcal{H}_{prop}$ orchestrates the correct gate propagation. The ground state is the “history state”

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

where $V$ prepares the input and $|t\rangle$ is the clock.

These constructions establish the formal equivalence, in computational power, between variational and standard gate-based quantum computation for arbitrary quantum circuits.

2. Optimization Process and Classical–Quantum Feedback

VQAs operate in a classical-quantum hybrid loop, where the parameters θ are updated by a classical optimizer based on quantum measurements of the chosen cost function. At each iteration, the quantum processor prepares the parameterized state, evaluates the cost (often as a sum of Pauli expectation values), and the classical processor updates θ targeting cost minimization. Notably, efficient optimization is facilitated when the structure of the objective function mirrors the circuit being simulated—the “optimization landscape" is then guided in a way that the optimal solution is efficiently accessible, akin to running the very circuit being simulated.

A key resource in this process is the number of distinct expected values (Pauli measurements) required to evaluate the cost function at each step. The cost of estimation of these expectation values directly impacts the practical feasibility of large-scale VQA runs.

3. Cost, Scaling, and Measurement Resources

Measurement overhead in VQA is governed primarily by the number of unique Pauli terms to be measured for evaluation of the cost function:

• Telescoping Construction: For an $n$-qubit circuit with $\mathcal{O}(\mathrm{poly}(\ln n))$ non-Clifford gates and any number of Clifford gates, the number of expected values to be measured is efficient and independent of the number of Clifford gates. Each non-Clifford gate can increase the number of Pauli terms, but Clifford gates preserve the cardinality of the Pauli decomposition—an invariance critical for scalability [“Clifford Gate Cardinality Invariance" lemma in (Biamonte, 2019)].
• History State (Feynman–Kitaev) Construction: For a circuit divided into L gate blocks, the measurement resource scales as $\mathcal{O}(L^2)$, with only $\mathcal{O}(\ln L)$ additional “slack” (clock) qubits required—a favorable trade-off for tracking long quantum circuits with manageable measurement and hardware overhead.

An explicit lower bound for the spectral gap of the constructed Hamiltonian is given (e.g., $\Delta \geq \max\{J, K\pi^2/2(L+1)^2 \}$), which is instrumental in bounding the variational error and establishing the necessary approximation guarantees.

4. Objective Function Design and Circuit Partitioning

Careful construction of the objective function and decomposition of the target circuit are central to the efficiency and universality of VQA:

• Partitioning the quantum circuit into L segments and encoding the computation with an efficient clock (requiring only $\mathcal{O}(\ln L)$ qubits) achieves universal objective functions with quadratic measurement scaling and logarithmic resource overhead in qubits. The ability to partition and efficiently track progression through circuit segments is critical for simulating long-depth circuits.
• In the telescoping scheme, efficient scaling requires limiting the growth in non-Clifford resources and exploiting the invariance property of Clifford gates to arrest exponential proliferation of Pauli measurements.

Such strategies collectively delineate a resource trade-off mapping: logarithmic scaling in ancilla (clock) qubits for polynomial measurement scaling, which is considered optimal for evolutionary simulations of complex quantum circuits in the variational model.

5. Implications for Universality and Practical Computation

The main consequence of these constructions is the formal establishment of VQA as a universal quantum computing model. Specifically:

• Any quantum algorithm expressed as a circuit over a universal gate set can be reformulated as a variational problem—i.e., as minimizing the expectation value of an appropriately engineered objective function with a classical-quantum feedback loop.
• This equivalence raises the status of VQAs: they are not solely heuristic tools for NISQ-era noisy devices, but are, in principle, capable of universal computation provided the resources (notably measurement overhead) are efficiently managed.
• For near-term quantum devices, particularly those limited by coherence time and error rates, VQAs offer a pathway to universal computation by leveraging shallow circuits, classical feedback, and flexible resource allocation between measurement and hardware.

Such universality establishes a rigorous bridge between hybrid-classical variational paradigms and traditional quantum models (gate-based, adiabatic, measurement-based), and provides a theoretical foundation for the future development and deployment of VQA-driven architectures.

6. Measurement Efficiency, Limitations, and Directions

The main technical challenges and limitations for VQA-based universal computation are:

• The number of required measurements can still become prohibitive for circuits with large numbers of non-Clifford gates or extremely long gate sequences, especially when the quadratic measurement scaling ($\mathcal{O}(L^2)$) becomes significant.
• While Clifford gates are measurement-efficient, non-Clifford gates drive up measurement resources. For practical quantum algorithms with substantial non-Clifford content, optimization of ansatz circuits and further reduction of measurement overhead remain pressing research areas.
• Effective error suppression or error-correction strategies must be integrated to realize full universality on hardware constrained by noise—this is especially relevant for NISQ-era devices.

Looking forward, the formal demonstration of universality for variational quantum computation motivates further exploration of resource-efficient objective constructions, scalable measurement reduction techniques, and error-mitigation schemes tailored to the variational paradigm.

7. Summary Table: Objective Function Constructions and Resource Scaling

Construction	Expected Value Scaling	Ancilla/Slack Qubits
Telescoping (penalty Hamiltonian)	$\mathcal{O}(\mathrm{poly}(\ln n))$ (if non-Clifford limited)	0
Feynman–Kitaev (history state)	$\mathcal{O}(L^2)$	$\mathcal{O}(\ln L)$

In summary, variational quantum algorithms provide a formally universal, resource-adaptable model for quantum computation, with rigorous constructions enabling the variational preparation of arbitrary quantum circuit outputs. These results shift VQAs from heuristic status to foundation for scalable quantum computation, and set the stage for sophisticated approaches to harnessing both quantum and classical processing in concert for complex problem solving (Biamonte, 2019).