A simplified version of the quantum OTOC$^{(2)}$ problem

Abstract: This note presents a simplified version of the OTOC$^{(2)}$ problem that was recently experimentally implemented by Google Quantum AI and collaborators. We present a formulation of the problem for growing input size and hope this spurs further theoretical work on the problem.

• The paper introduces a simplified quantum OTOC(2) formulation, enabling tractable expectation value estimation on random quantum circuits.
• It presents efficient quantum algorithms with O(nd) gate complexity and emphasizes classical simulation hardness through inverse polynomial fluctuations and sign problems.
• The work establishes a practical benchmark for quantum advantage, focusing on verifiability and implications for quantum simulation and complexity theory.

Simplified Quantum OTOC$^{(2)}$ Problem: Formulation and Complexity

Introduction

This paper introduces a streamlined version of the quantum OTOC$^{(2)}$ (Out-of-Time-Order Correlator) problem, motivated by recent experimental implementations by Google Quantum AI. The work aims to provide a theoretically tractable formulation for growing input sizes, facilitating further analysis of quantum advantage in expectation value problems. The focus is on function computation tasks, which, unlike sampling problems, yield verifiable outputs for each input instance, albeit not always classically verifiable.

Problem Formulation

The core problem is defined on a random quantum circuit $U$ acting on $n = \ell \times \ell$ qubits arranged in a 2D grid. The circuit ensemble consists of Haar-random 2-qubit gates in a brickwork pattern of four layers, alternating between horizontal and vertical arrangements. The observable of interest is constructed as follows:

• $B$: Pauli $X$ operator on qubit $(\ell, \ell)$ (the "butterfly" operator)
• $M$: Pauli $Z$ operator on qubit $(1, 1)$ (the "measurement" operator)
• $C = U^\dagger B U M$

The task is to estimate $\langle 0^n | C^4 | 0^n \rangle$ to additive error $\epsilon$. For shallow circuits, $C^4$ reduces to the identity, yielding an expectation value of $1$. As circuit depth increases and operator commutation is lost, $C^4$ becomes scrambled, and the expectation value approaches $0$. The transition regime is conjectured to exhibit inverse polynomial fluctuations across instances.

Quantum and Classical Complexity

Quantum algorithms can estimate $\langle 0^n | C^4 | 0^n \rangle$ efficiently:

• Gate complexity: $O(nd)$ for circuit implementation
• Sample complexity: $O(1/\epsilon^2)$ for additive error $\epsilon$ (amplitude estimation can reduce this to $O(1/\epsilon)$ for deeper circuits)

The paper conjectures classical hardness for large $n$, specifically that no classical algorithm exists with complexity $\mathrm{poly}(n, d, 1/\epsilon)$ for certain parameter regimes (e.g., $d \in \Theta(\ell)$, $\epsilon = 1/\mathrm{poly}(n)$). This is supported by evidence from recent experiments and theoretical arguments regarding the sign problem in classical Monte Carlo methods, which worsens with higher moments $k$ in $\langle 0^n | C^{2k} | 0^n \rangle$.

Generalizations and Theoretical Implications

The framework extends naturally to higher moments, $\langle 0^n | C^{2k} | 0^n \rangle$, and to expectation values on the maximally mixed state, $\mathrm{Tr}(C^{2k})/2^n$. These generalizations quantify deviations from $2k$-designs and further exacerbate classical simulation difficulties due to sign problems. The simplified problem formulation abstracts away hardware-specific details, focusing on the essential computational complexity and verifiability aspects.

Experimental Comparison

The experimental implementation diverges from the theoretical model in several respects:

• Qubit layout: $n=65$ qubits, not a perfect square grid
• Circuit depth: $d=23$ layers
• Operators: $B$ is a 3-qubit Pauli $X$, $M$ is Pauli $Z$
• Gate set: Fixed 2-qubit gates (iSWAP-like) and random single-qubit gates
• Measured quantity: A more complex variant of $\langle 0^n | C^4 | 0^n \rangle$, with the diagonal part subtracted
• Error metric: Signal-to-noise ratio (Pearson correlation), not uniform additive error

The paper posits that achieving additive error $\epsilon = 0.001$ would be a significant challenge for classical algorithms in this regime.

Practical and Theoretical Implications

The simplified OTOC$^{(2)}$ problem provides a concrete benchmark for quantum advantage in expectation value estimation, with implications for quantum simulation and verification. The conjectured classical hardness, especially for higher moments and large system sizes, suggests that quantum devices can outperform classical algorithms in specific, verifiable tasks. This has direct relevance for quantum simulation of physical systems, where quantum verifiability is sufficient and classical verification is infeasible.

The abstraction of hardware details enables broader theoretical study, potentially leading to new complexity-theoretic insights and improved quantum algorithms. The sign problem in classical simulation remains a central barrier, and further research may elucidate its role in delineating quantum-classical boundaries.

Conclusion

The paper presents a theoretically motivated, simplified version of the quantum OTOC$^{(2)}$ problem, highlighting its suitability as a benchmark for quantum advantage in expectation value estimation. The formulation abstracts away experimental specifics, focusing on computational complexity and verifiability. The conjectured classical hardness and efficient quantum algorithms underscore the potential for quantum devices to solve problems beyond classical reach, with significant implications for quantum simulation and complexity theory. Future work may explore higher moments, alternative circuit ensembles, and refined verification protocols to further characterize the quantum-classical divide.