[Paper Review] Limitations of variational quantum algorithms: a quantum optimal transport approach

Giacomo De Palma, Milad Marvian+2 more|arXiv (Cornell University)|Apr 7, 2022

Quantum Computing Algorithms and Architecture94 references83 citations

TL;DR

This paper establishes tight, exponentially small bounds on the performance of variational quantum algorithms—such as QAOA and quantum annealing—under both noisy and noiseless conditions using a novel quantum optimal transport framework. It proves that for Max-Cut problems on D-regular graphs, QAOA requires depth L ≥ (1/2) log(D+1) log n / 576 to outperform classical algorithms, and that with local depolarizing noise p, performance degrades exponentially at depths L = O(p⁻¹), making success exponentially unlikely. The work introduces new quantum entropic and concentration inequalities rooted in optimal transport theory, offering a unified toolkit applicable beyond variational algorithms.

ABSTRACT

The impressive progress in quantum hardware in the last years has raised the interest of the quantum computing community in harvesting the computational power of such devices. However, in the absence of error correction, these devices can only reliably implement very shallow circuits or comparatively deeper circuits at the expense of a nontrivial density of errors. In this work, we obtain extremely tight limitation bounds for standard NISQ proposals in both the noisy and noiseless regimes, with or without error-mitigation tools. The bounds limit the performance of both circuit model algorithms, such as QAOA, and also continuous-time algorithms, such as quantum annealing. In the noisy regime with local depolarizing noise $p$, we prove that at depths $L=\mathcal{O}(p^{-1})$ it is exponentially unlikely that the outcome of a noisy quantum circuit outperforms efficient classical algorithms for combinatorial optimization problems like Max-Cut. Although previous results already showed that classical algorithms outperform noisy quantum circuits at constant depth, these results only held for the expectation value of the output. Our results are based on newly developed quantum entropic and concentration inequalities, which constitute a homogeneous toolkit of theoretical methods from the quantum theory of optimal mass transport whose potential usefulness goes beyond the study of variational quantum algorithms.

Motivation & Objective

• To establish rigorous, tight limitations on the performance of variational quantum algorithms in both noisy and noiseless regimes.
• To develop a new theoretical toolkit based on quantum optimal transport for analyzing shallow and noisy quantum circuits.
• To demonstrate that error mitigation protocols fail to reverse performance degradation under realistic noise and depth constraints.
• To prove that classical algorithms outperform noisy quantum circuits not only in expectation but with exponentially small probability of better outcomes.

Proposed method

• Derives new quantum entropic and concentration inequalities using the (2, ∞)-Poincaré inequality from quantum optimal transport theory.
• Applies Milman’s optimal transport inequality to analyze the concentration of measure for output states of shallow quantum circuits on the Hamming space {0,1}^n.
• Models noisy circuits with layers of one-qubit depolarizing noise and derives bounds on the probability of observing high-energy outcomes.
• Uses a dilation-based approach to control the Lipschitz constant of observables under unitary evolution, enabling variance bounds via the Poincaré inequality.
• Analyzes error mitigation protocols such as virtual distillation and expectation estimation, showing their failure at constant or logarithmic depth.
• Applies duality and trace norm estimates from Wasserstein distance formulations to bound the contraction of quantum states under continuous-time evolution.

Experimental results

Research questions

• RQ1 What is the minimum circuit depth required for QAOA to outperform classical algorithms on Max-Cut for D-regular graphs?
• RQ2 How does local depolarizing noise affect the probability of observing a quantum advantage in variational algorithms?
• RQ3 Can error mitigation techniques reverse performance degradation in shallow, noisy quantum circuits?
• RQ4 To what extent do quantum optimal transport tools unify and refine existing bounds on quantum circuit output distributions?
• RQ5 What is the concentration profile of the output measure of noisy quantum circuits at finite depth?

Key findings

• For Max-Cut on D-regular graphs, QAOA requires depth L ≥ (1/2) log(D+1) log n / 576 to outperform classical algorithms, which is an exponential improvement over prior bounds in D.
• At depths L = O(p⁻¹) with local depolarizing noise p, the probability that a noisy quantum circuit outperforms classical algorithms is exponentially small in the number of qubits.
• Even with error mitigation, protocols like virtual distillation have exponentially small success probability at constant depth, and expectation estimation fails at O(log n) depth.
• The paper proves that m copies of a noisy quantum circuit provide no significant advantage over m copies of a trivial product state when m = poly(n), under error mitigation.
• The derived quantum optimal transport toolkit provides a homogeneous framework that unifies and strengthens prior results on concentration and variance in quantum circuits.