Quantum-Classical Separation in Bounded-Resource Tasks Arising from Measurement Contextuality

TL;DR

The paper experimentally probes quantum contextuality as a bounded-resource computational resource using a superconducting-qubit processor. It demonstrates contextual advantages through the Mermin-Peres magic square, Kochen-Specker-Bell inequalities, and scalable GHZ parity games up to 71 qubits, followed by a benchmark on the 2D hidden linear function problem with shallow circuits. By decomposing fidelity into T1, gate, and readout error channels and introducing a contextuality-based benchmarking framework, the work shows how contextuality can serve as a practical metric for hardware performance and a resource for bounded-resource quantum tasks. Together, these results highlight the potential of contextuality-aware tasks for hardware benchmarking and for exploring the limits of near-term quantum devices.

Abstract

The prevailing view is that quantum phenomena can be harnessed to tackle certain problems beyond the reach of classical approaches. Quantifying this capability as a quantum-classical separation and demonstrating it on current quantum processors has remained elusive. Using a superconducting qubit processor, we show that quantum contextuality enables certain tasks to be performed with success probabilities beyond classical limits. With a few qubits, we illustrate quantum contextuality with the magic square game, as well as quantify it through a Kochen--Specker--Bell inequality violation. To examine many-body contextuality, we implement the N-player GHZ game and separately solve a 2D hidden linear function problem, exceeding classical success rate in both. Our work proposes novel ways to benchmark quantum processors using contextuality-based algorithms.

Figures (14)

• Figure 1: The Mermin-Peres magic square game.(A) The rules of this game. The green dashed line indicates the incommunicado constraint. Alice gets a row and Bob a column of a $3 \times 3$ table; entries must satisfy row products $= +1$, column products $= -1$, and they win if their shared cell matches. (B) Leveraging a quantum strategy, the players share two Bell pairs and agree on what measurements to perform for each assigned row (Alice) or column (Bob), using the measurement plan table, where first Pauli in each cell refers to their first qubit. (C) Each player performs two simultaneous measurements and, given the product constraint rule (iii), they infer the third value needed to fill out their assigned row or column. (D) Measured probability of winning the game when played on our processor, shown for 9 distinct games (row or column assignment) sorted by where the assigned rows and columns intersect. The average probability of winning is $P_w=0.9830 \pm 0.0001$. Each game is played $N_s=100{,}000$ times and the error bar plotted for each game is 100 times the statistical uncertainty derived from the binomial distribution.
• Figure 2: Quantifying contextuality by measuring $\chi_{\text{KSB}}$.(A) Measuring $\chi_{\text{KSB}}$ requires successive quantum non-demolition (QND) measurements of two qubits (Q1, Q2) which we achieve by using an ancilla qubit (An). We construct a long circuit of 180 randomly selected rows or columns (contexts, 540 QND measurements) and run the sequence for $1{,}000$ times. Using single qubit and two qubit entangling gates, a random sequence of rows and columns of Table \ref{['tab:example_table']} is measured and after each measurement the ancilla is reset to the $|0\rangle$ state. (B) Measuring each row or column (a "context") entails three successive measurements, each $\pm 1$, which in the absence of noise, would multiply to +1 for the rows or $-1$ for the columns. Plotted are $6\langle R_i\rangle$ and $-6\langle C_i\rangle$, for all rows and columns; error bars indicate 10 $\times$ the statistical uncertainty, $\sigma$. The horizontal dashed line corresponding to $\chi_{\rm KSB}=4$ is the classical limit assuming pre-existing classical values of $\pm 1$ for each of the 9 elements of the table and the olive, horizontal solid line corresponds to the sums of the averages $\chi_{\text{KSB}} = 5.618 \pm 0.005$, as given in Eq. \ref{['eq:chi']}. Additional data are provided in the Supplementary Materials.
• Figure 3: The $N$-qubit GHZ parity game.(A) Quantum circuit for the game. Each of the $N$ players is given a binary value $x_j$ and is asked to return a binary answer $y_j$ such that the sum of the answers (mod 2) is the same as the sum of the questions divided by two. Using an $N$-qubit GHZ state, players can answer the questions by measuring in a corresponding basis, depending on the binary question ($x_j$) they were assigned. (B) The average winning probability measured on up to 71 qubits (or players). 20 distinct games (a random set of $x_j$ values) were selected, and each game was implemented on 50 choices of GHZ state growth pattern. Each of these 1000 configurations was measured $N_s=1000$ times. The shaded regions are $\pm 20 \sigma$ and $\pm 40 \sigma$ of statistical error computed from the 50 qubit configurations used, after averaging over 20 games played (see Supplementary Materials).
• Figure 4: Quantum advantage in finding solutions of 2D hidden linear functions. (A) Quantum circuit for finding a hidden function. Qubits are initialized in a product $|0\rangle$ state and then a gate is applied to all of them. The encoding of the symmetric $n\times n$ matrix $A$ is done by applying an $S$-gate where $A_{i,i}$ is non-zero and CZ-gate between qubits $i$ and $j$ where $A_{i,j}$ is nonzero. The diagonal and allowed off-diagonal elements in $A$ are non-zero with probability 0.5, where the allowed off-diagonal elements are those which connect neighboring qubits. At the end all qubits are measured in the computational basis, after a layer of Hadamard gates on all qubits. (B) The effective number of layers is defined as 4 divided by the fraction of total number of measured bitstrings that are correct (right axis). Solid data points are the average of $N_g=1000$ distinct matrices, each run $N_s=100$ times, at each $n$-value, and the standard deviation $\sigma$ is the statistical fluctuations of $N_g$ distinct runs (see Supplementary Materials for details). The gray horizontal line segments are the classical lower bound.
• Figure S1: Representative error rates for a 71-qubit 2D grid on the 72-qubit Willow processor and a 105-qubit 2D grid on the 105-qubit Willow processor. The data include readout (measurement) errors, single-qubit randomized benchmarking (1Q RB) errors, and two-qubit cross-entropy benchmarking (2Q XEB) Pauli errors for CZ gates. Vertical dashed lines denote the median values for each dataset. The readout errors $\mathbf{e_0}$ and $\mathbf{e_1}$ correspond to the probability of reporting $|1\rangle$ when the qubit is prepared in $|0\rangle$ and of reporting $|0\rangle$ when prepared in $|1\rangle$, respectively.