Heuristic Quantum Advantage with Peaked Circuits

TL;DR

The paper presents HQAP circuits as a scalable, verifiable route to heuristic quantum advantage, demonstrated on Quantinuum's System Model H2 with up to 56 qubits and 2044 two-qubit gates, where quantum runtimes are orders of magnitude shorter than those projected for classical simulators. It details a robust protocol for constructing peaked circuits using a shallow random circuit followed by a variational peaking layer, plus identity-obfuscation, tensor patch optimization, and swap-based scrambling to impede classical contraction paths. The authors provide comprehensive benchmarking against state-of-the-art classical methods (MPS, TN+BP, PPS), showing an empirical gap that grows with circuit size and depth, and they prove that deciding peakedness in general is QCMA-hard, motivating a potential post-quantum encryption application. The work frames the observed gap as a practical, testable form of quantum advantage, distinct from unconditional complexity proofs, and it invites community participation via public challenges to further delineate classical limitations. Overall, HQAP circuits offer a concrete, verifiable benchmark for utility-scale quantum hardware and a pathway toward quantum-safe cryptographic proposals, while highlighting the ongoing tension between empirical demonstrations and formal hardness results.

Abstract

We design and demonstrate heuristic quantum advantage with peaked circuits (HQAP circuits) on Quantinuum's System Model H2 quantum processor. Through extensive experimentation with state-of-the-art classical simulation strategies, we identify a clear gap between classical and quantum runtimes. Our largest instance involves all-to-all connectivity with 2000 two-qubit gates, which H2 can produce the target peaked bitstring directly in under 2 hours. Our extrapolations from leading classical simulation techniques such as tensor networks with belief propagation and Pauli path simulators indicate the same instance would take years on exascale systems (Frontier, Summit), suggesting a potentially exponential separation. This work marks an important milestone toward verifiable quantum advantage, as well as providing a useful benchmarking protocol for current utility-scale quantum hardware. We sketch our protocol for designing these circuits and provide extensive numerical results leading to our extrapolation estimates. Separate from our constructed HQAP circuits, we prove hardness on a decision problem involving generic peaked circuits. When both the input and output bitstrings of a peaked circuit are unknown, determining whether the circuit is peaked constitutes a QCMA-complete problem, meaning the problem remains hard even for a quantum polynomial-time machine under commonly accepted complexity assumptions. Inspired by this observation, we propose an application of the peaked circuits as a potentially quantum-safe encryption scheme~\cite{chen2016report,kumar2020post,joseph2022transitioning,dam2023survey}. We make our peaked circuits publicly available and invite the community to try additional methods to solve these circuits to see if this gap persists even with novel classical techniques.

Figures (11)

• Figure 1: Gap in Classical and Quantum Runtimes for Solving 56 qubit HQAP Circuits. a) We compare classical techniques (MPS, PPS and TNS+BP) with quantum techniques for solving 56 qubit, 10% peaked circuits with all-to-all connectivity. As number of RZZs increases, classical methods require estimated millions of Nvidia H100 GPU hours, while the quantum device requires less than 2 hours. More details on the classical methods can be found in section \ref{['sec:classical_methods']}b) The performance of the quantum device drops with the growing number of RZZs in the HQAP circuit as expected, with still strong signal of 5/1000 (correct) peak samples for a 2044 RZZ circuit. c) Actual results from a single run on Quantinuum H2 for a 1917 RZZ peaked circuit, with 17/2000 correct peaked bitstrings. To our knowledge no other device in the world can find the peak for HQAP circuits of this scale and connectivity.
• Figure 2: Visual schematic of HQAP building. We include details of the circuit manipulations used to scramble the circuit structure.
• Figure 3: Estimating $\chi_{\rm break}$. By tracking the accuracy $R$ as a function of $\chi$, we find 3 cases. The black curve is an example of a circuit where $\chi_{\rm break}$ is found exactly. For the blue curve, the $\chi_{\rm break}$ is not found exactly, and so we fit the data to find $\hat{R}(\chi)$ and extrapolate to estimate $\chi_{\rm break}$, and show the fit with a dashed line. Lastly, the gray curve is an example of a circuit our fitting protocol rejects when estimating $\chi_{\rm break}$.
• Figure 4: Benchmarking of bond dimension and time required to simulate peaked circuits of increasing depth with MPS. (a) Bond dimension vs two-qubit gate count. Black dots indicate MPS simulations that converged exactly, while white dots indicate circuit samples where the extrapolation procedure in Fig. \ref{['fig:mps_chi_scaling']} was used. Blue dots indicate the transition where some circuits of that size were fully solved. The red dashed line shows the exponential fit, while the solid purple line shows the $\chi$ corresponding to exact simulation. (b) Gate simulation cost for increasing $\chi$. We study how the simulation time grows with increasing bond dimension, normalized by the number of gates. The large $\chi$ behavior is modeled by by a cubic function of $\chi$. (c) We plot the solution time $T_{\rm break}$ in hours as a function of circuit depth. Exact answers (black dots) can be read off, while white dots are extrapolated by using $\hat{\chi}_{\rm break}$ and the time conversion in (b). The red dashed line is an exponential fit for $T_{\rm break}$ compared with circuit depth. The purple dashed line is the estimate for the runtime of a depth 2000 circuit with $\chi_{\rm sat}$.
• Figure 5: Example of the tree-like graph that we transpile our all-to-all circuits to before performing TNS simulations. The lack of loops lead to confidence in the convergence of BP for computing marginals in the tensor network state.

Theorems & Definitions (7)

• Definition 5.1: Peakedness check on basis states (PCBS)
• Definition 5.2: QCMA
• Theorem 5.3: PCBS is QCMA-complete
• Theorem 5.4
• proof : Proof
• Corollary 5.5
• Conjecture 5.6: Peak-Search Hardness (PSH)