Simplified Quantum OTOC(2) Benchmark

• The topic is a simplified, experimentally tractable version of higher-moment out-of-time-order correlators that captures quantum scrambling via fourth moment measurements.
• It benchmarks quantum circuits by analyzing operator spreading and non-commutativity techniques, highlighting regions where classical simulation becomes intractable.
• The problem offers practical insights into quantum chaos with scalable protocols using random circuit designs and amplitude estimation to enhance measurement efficiency.

A simplified quantum OTOC$^{(2)}$ problem is a computationally and experimentally tractable version of the higher-moment out-of-time-order correlator task, designed as a benchmark for quantum circuits and as a platform for theoretical complexity analysis. It is formulated to capture the essential features of quantum information scrambling via OTOC moments, while removing some of the physical and algorithmic overhead of more general OTOC measurement schemes. The task crystallizes advances in random circuit protocols and leverages both quantum circuit sampling and measurement to probe the onset of non-commutativity, sensitivity to circuit depth, and the transition to hard-to-simulate regimes. The development of this problem is motivated in part by recent experimental efforts, including those by Google Quantum AI, to establish practical quantum advantage using correlation observables beyond standard circuit sampling.

1. Formal Definition and Circuit Structure

The core problem is stated as follows: Given a 2D array of $n = \ell \times \ell$ qubits, consider a quantum circuit $U$ built from $d$ layers of two-qubit Haar-random gates in a fixed brickwork pattern alternating between horizontal and vertical arrangements. Two operator observables are specified:

• A "butterfly operator" $B$ (e.g., a single-qubit Pauli $X$ acting on the far corner, such as $(\ell, \ell)$)
• A "measurement operator" $M$ (e.g., a Pauli $Z$ acting on the opposite corner, such as $(1,1)$)

Define the composite correlation operator as $C = U^{\dagger} B U M$. The central computational task is to estimate the fourth moment (the second-order OTOC)

$\langle 0^n | C^4 | 0^n \rangle$

where $|0^n\rangle$ is the all-zero computational basis state. This moment amplifies the signal in the scrambling regime, where $C^2$ is nearly traceless due to operator spreading and non-commutativity.

Notably, in shallow circuits ($d$ small), $U^{\dagger} B U$ and $M$ commute and the correlation is trivial; in deeper circuits, the operators' lightcones overlap, leading to non-commutation and $\langle 0^n|C^4|0^n\rangle$ becomes exponentially small—a regime intractable by direct classical simulation.

The quantum algorithm for the problem prepares $|0^n\rangle$, applies $C^2$, and measures $M$ to accumulate statistics for the fourth moment to a prescribed additive error $\epsilon$. Algorithmic gate complexity scales as $O(nd)$ in the depth $d$ and system size $n$, with a dependence on $\epsilon$ controlled by measurement repetitions; amplitude estimation can, in principle, quadratically improve sampling efficiency.

2. Experimental Realizations and Variants

Recent large-scale superconducting qubit experiments have demonstrated the practical feasibility of this protocol. For instance, implementations used $n=65$ qubits with circuit depths up to $d=23$. Practical alterations included:

• Employing multi-qubit Pauli strings for $B$ and $M$
• Realizing non-Haar ensembles via fixed $i$SWAP-like two-qubit gates composed with random single-qubit rotations, mimicking the typical circuit structure but facilitating hardware convenience
• Adjusting the output statistic: the reported quantity in the experiment subtracted a "diagonal component" (denoted $\mathcal{C}^{(4)}_{\rm diag}$), isolating the off-diagonal quantum interference responsible for classical hardness and sensitivity.

Crucially, measurement error was evaluated not by additive error per se, but via signal-to-noise ratio (SNR) between ideal and experimental outcomes—a metric that emphasizes the deep classical hardness for even modest values (e.g., SNR$\sim$5 is already challenging for state-of-the-art classical algorithms at $n\sim40$).

3. Scaling and Complexity as Input Size Grows

The problem scales naturally with input size: for a system of $n = \ell \times \ell$ qubits and $d \sim \ell$, the quantum algorithm continues to operate efficiently (gate depth and qubit connectivity permitting). Theoretical conjectures in the paper suggest that for $d \in \Theta(\ell)$ and small $\epsilon = 1/\mathrm{poly}(n)$, the classical simulation of $\langle 0^n|C^4|0^n\rangle$ is hard (even average-case hard), paralleling Random Circuit Sampling in its computational complexity landscape. Furthermore, the protocol is flexible—alternative moments $\langle 0^n|C^{2k}|0^n\rangle$ for $k > 2$ can in principle be considered to interpolate between regimes of trivial and fully random correlations.

System size $(n)$	Circuit depth $(d)$	Quantum gate complexity	Classical simulation SNR
$O(100)$	$O(20-30)$	$O(nd)$	$\sim 1$ (classically hard)

Above: Scaling regime for which the task is expected to be quantum-tractable and classically hard (as demonstrated in recent experiments).

4. Foundations in Out-of-Time-Order Correlators

The motivation for using the fourth moment $\langle 0^n|C^4|0^n\rangle$ is rooted in the properties of OTOCs as probes of quantum chaos and scrambling. As $U$ becomes deep, operator evolution (Heisenberg picture) causes $B$ to spread over many Pauli strings; nonzero expectation values arise only from highly nontrivial operator overlaps with $M$. For $C^2$, the overlap signal gets exponentially small, whereas the fourth moment retains detectable structure and is more robust against experimental and statistical noise. This design explicitly targets the simulated regime where conventional OTOC-based chaos detection becomes computationally intractable for classical hardware, but remains operationally feasible for quantum processors.

5. Justification for Quantum Hardness

The classical hardness of the task follows from the structure of the circuit and output observable. In the deep circuit regime, the measurement amounts to evaluating a sum with exponentially many terms, generically afflicted by the "sign problem": highly oscillatory contributions with rapidly fluctuating phases and no efficient importance sampling scheme. The use of higher OTOC moments such as $C^4$ worsens this sign problem, and there is evidence (from both classical simulations and the quantum experiments referenced) that even the best known hybrid tensor network and Monte Carlo methods are inefficient in this regime.

A plausible implication is that since even modest error in the measurement of $\langle 0^n|C^4|0^n\rangle$ exceeds what is accessible classically for $n \gtrsim 40$ and $d \gtrsim 20$, this problem is a promising platform for demonstrating average-case quantum advantage in a physically meaningful observable.

6. Directions for Theory and Further Study

The paper highlights several avenues for future progress:

• Rigorous complexity-theoretic analysis of the classical hardness of the simplified OTOC$^{(2)}$ problem under realistic circuit and noise assumptions
• Understanding the crossover as a function of circuit depth from easy (commuting) to hard (fully scrambled) regimes and its relation to random matrix theory predictions for OTOC moments
• Extending the protocol to higher moments and understanding the emergence of classical hardness as $k$ increases in $\langle 0^n|C^{2k}|0^n\rangle$
• Exploring connections to quantum verification schemes and cross-platform benchmarking, as the observable lends itself naturally to settings where two quantum devices can be used to verify each other's measurement outcomes for the same instance.

7. Summary Table: Core Components

Component	Description
Circuit ensemble	Brickwork of Haar-random 2-qubit gates on 2D lattice
Operators ($B$, $M$)	Local Pauli operators at distant sites
Central quantity	$\langle 0^n | (U^{\dagger}BU M)^4 | 0^n\rangle$
Quantum complexity	$O(nd/\epsilon^2)$ (improvable to $O(nd/\epsilon)$ with amplitude estimation)
Classical intractability	Exponential runtime, with SNR rapidly decaying as $n,d$ grow

In essence, the simplified quantum OTOC$^{(2)}$ problem distills the physics of operator spreading and quantum scrambling into a minimal, scalable computational benchmark, simultaneously serving experimental, theoretical, and complexity-theoretic research objectives. It provides a clean context for the analysis of OTOC-based quantum advantage and is directly motivated by the operational demands and technical lessons of contemporary quantum hardware (King et al., 22 Oct 2025).