Limitations of Linear Cross-Entropy as a Measure for Quantum Advantage

TL;DR

This work critically evaluates Linear Cross-Entropy Benchmark (XEB) as a benchmark for quantum advantage, showing that XEB's fidelity proxy is fragile under adversarial settings. By mapping average XEB and fidelity to a diffusion-reaction Markov process via unitary 2-design properties, the authors derive a transfer-matrix framework with diffusion rate $D$ and reaction rate $R$ that explains scrambling, noise effects, and stationary behavior in both 1D and 2D circuit architectures. They demonstrate that high XEB can be achieved without faithful quantum dynamics by exploiting additive XEB contributions and gate omissions, and they introduce improved strategies (mixed-state simulation, MDN, and top-$k$ sampling) to robustly spoof XEB in certain regimes. The work also provides a pathway to refute XQUATH for multiple architectures and discusses how to reinterpret XEB through Ising-model mappings, offering practical insights into the limitations and potential improvements of XEB as a benchmark for quantum advantage.

Abstract

Demonstrating quantum advantage requires experimental implementation of a computational task that is hard to achieve using state-of-the-art classical systems. One approach is to perform sampling from a probability distribution associated with a class of highly entangled many-body wavefunctions. It has been suggested that this approach can be certified with the Linear Cross-Entropy Benchmark (XEB). We critically examine this notion. First, in a "benign" setting where an honest implementation of noisy quantum circuits is assumed, we characterize the conditions under which the XEB approximates the fidelity. Second, in an "adversarial" setting where all possible classical algorithms are considered for comparison, we show that achieving relatively high XEB values does not imply faithful simulation of quantum dynamics. We present an efficient classical algorithm that, with 1 GPU within 2s, yields high XEB values, namely 2-12% of those obtained in experiments. By identifying and exploiting several vulnerabilities of the XEB, we achieve high XEB values without full simulation of quantum circuits. Remarkably, our algorithm features better scaling with the system size than noisy quantum devices for commonly studied random circuit ensembles. To quantitatively explain the success of our algorithm and the limitations of the XEB, we use a theoretical framework in which the average XEB and fidelity are mapped to statistical models. We illustrate the relation between the XEB and the fidelity for quantum circuits in various architectures, with different gate choices, and in the presence of noise. Our results show that XEB's utility as a proxy for fidelity hinges on several conditions, which must be checked in the benign setting but cannot be assumed in the adversarial setting. Thus, the XEB alone has limited utility as a benchmark for quantum advantage. We discuss ways to overcome these limitations.

Figures (16)

• Figure S1: Different types of error contributions. (a) Error cancellations. (b) One large light-cone. (c) Overlap of two light-cones. (d) Multiple small light-cones.
• Figure S2: Additivity of the XEB vs. multiplicativity of the fidelity. After changing the reference frame, we consider two isolated quantum circuits as the "ideal" reference point.
• Figure S3: A large discrepancy between the XEB and the fidelity for deep quantum circuits. (a) Top. We use the $k$-th moments of the bitstring output distribution as an indicator of the circuit being in the scrambling regime. Concretely, we plot the ratio between the $k$-th moments of the output distribution and that of the Porter-Thomas (PT) distribution for circuits with 24 qubits and the 2-qubit Haar random gates. The approximate PT distribution can be seen explicitly (inset of Bottom). The numerical result suggests that the circuit is sufficiently scrambling when the depth is greater than 20, indicated by "scrambling enough". Bottom. XEB and fidelity as a function of circuit depth $d$ for noisy circuit and for our algorithm. In both cases, the XEB and the fidelity decay exponentially in depth. For noisy circuits, the XEB and fidelity values stay reasonably close to one another. For our algorithm, however, the XEB and the fidelity deviate from one another as they exponentially decay with different rates. (b) The average XEB and fidelity of circuits with omitted gates in the middle (i.e., our algorithm) are the same as that of circuits with independent random gates.
• Figure S4: Diagram of averaging over Haar ensemble. The blue box represents a single qubit Haar random unitary $u$ and the blue box with a $^*$ represents $u^*$. (a) Averaging for $t=1$ copy (i.e., unitary 1-design property). The diagram can be understood as either a tensor network representation or a circuit (non-unitary after averaging) of a state in Choi representation in which $\text{Tr}\rho_1=\langle\langle00|\rho_1\rangle\rangle+\langle\langle11|\rho_1\rangle\rangle$ . (b) Diagram of $|I\rangle\rangle$ and $|\Omega\rangle\rangle$ which are defined in Eq. \ref{['Seq:I_Omega']}. (c) Averaging for $t=2$ copies (i.e., unitary 2-design property) which is a diagram of Eq. \ref{['eq:app_2design']} by using (b).
• Figure S5: Illustration of the mapping from the average XEB of the ideal circuit to the diffusion-reaction model. (a) The XEB of ideal circuit can be computed by considering two copies of a state evolving under the same random quantum circuit. Gray dashed boxes defines simple representations of tensors that appear in our tensor network diagrams. The first box gives a shortcut of the diagram for the average behavior of each single qubit gate as discussed in Fig. \ref{['fig:12design']}(c) and Eq. \ref{['eq:app_2design']}. A circle is labelled by a classical variable $s\in \{I, \Omega\}$ and represents the corresponding "4-qubits states" (or vectorized density matrix in duplicated Hilbert space) $|I\rangle\rangle$ and $|\Omega\rangle\rangle$ defined in Fig. \ref{['fig:12design']}(b) and Eq. \ref{['eq:app_2design']} (denoted as 4 lines). The $W$ is a diagonal matrix defined for the classical degree of freedom gives. The second box defines a simplified diagram for the four copies of a 2-qubit entangling gate. (b) The tensor network of the diffusion-reaction model where the horizontal direction is viewed as time evolution of a Markovian process, described by $\mathcal{T}_1,\cdots,\mathcal{T}_d$, on the classical degree of freedom. In each $\mathcal{T}_i$, a matrix $T_0$ (where we omitted the gate dependence $^{(G)}$ here displayed in main text for $T_0^{(G)}$) on the classical degree of freedom is defined as the two copies of 2-qubit entangling gate combined with the white circles. Then combining $T_0$ with $W$, we get the transfer matrix $T$ which is a Markovian (will be proved in subsection \ref{['sapp:DR_transfer']}). In (a), there are two sets of independent single qubit Haar random gates on a wire between two successive entangling gates. They can be merged into one because the product of two independent Haar random unitary gates equals a single Haar random unitary gate. This is why we only have one layer of $W$'s between two successive layers of entangling unitary gates in (b). Here we only present the tensor network diagram for XEB, fidelity only differs at the right boundary condition as discussed in Fig.8(a) and (b) of the main text.

Theorems & Definitions (1)

• Conjecture 1: Linear Cross-Entropy Quantum Threshold Assumption (XQUATH) aaronson2019classical