Simon's algorithm in the NISQ cloud

TL;DR

The paper benchmarks Simon's algorithm on six NISQ devices accessed via the cloud to evaluate quantum advantage under realistic noise. It uses two oracle variants and reports the algorithmic error rate across problem sizes, contrasting hardware results with noisy simulators. Findings show error rates grow with problem size and approach random guessing (~$50\%$) for larger instances, with IonQ devices exhibiting roughly linear scaling and IBM devices showing topology-driven non-linear effects due to spatially distant two-qubit gates. The work underscores that current NISQ hardware, especially when accounting for chip topology and transpilation, is unlikely to sustain Simon's algorithm's quantum advantage, and highlights the need for architecture-aware compilation and further study of error-tolerant hidden-subgroup computations.

Abstract

Simon's algorithm was one of the first problems to demonstrate a genuine quantum advantage. The algorithm, however, assumes access to noise-free qubits. In our work we use Simon's algorithm to benchmark the error rates of devices currently available in the "quantum cloud." As a main result we obtain an objective comparison between the different physical platforms made available by IBM and IonQ. Our study highlights the importance of understanding the device architectures and chip topologies when transpiling quantum algorithms onto hardware. For instance, we demonstrate that two-qubit operations on spatially separated qubits on superconducting chips should be avoided.

Figures (10)

• Figure 1: Quantum circuit diagram for Simon's algorithm: Note the use of two quantum registers of size $n$, both initialized to the zero state $|0\rangle$, as well as the oracle $U_f$. The algorithm uses two Hadamard transformations and $U_f$ to create a superposition over all the size $n$ bitstrings that are orthogonal to the secret string $s$. An iteration of the algorithm is successful if the final measurement result is indeed orthogonal to $s$ (this is always true in the noise free case). The entire algorithm is successful if a complete set of $n-1$ linearly independent bitstrings are measured and the resulting system is solved classically in polynomial-time.
• Figure 2: Device topologies for IBM computers: The color of qubits and connections represent the single qubit readout error and ECR error, respectively, at the time the image was taken. Lighter colors correspond to higher error rates. All images retrieved from https://quantum.ibm.com/composer.
• Figure 3: Device topologies for IBM computers: Observe that in trapped ion computers we have full all-to-all connectivity. All images retrieved from https://us-west-2.console.aws.amazon.com/braket/home?region=us-west-2#/devices.
• Figure 4: Complex and simple oracle for Simon's algorithm. Each dotted wire represents $(n-2)$ qubits, and all operations on dotted wires are repeated following the pattern across all qubits.
• Figure 5: Simon's algorithm performed for the complex oracle at various sizes on IBM devices and simulators, cf. Fig. \ref{['fig:oracle']}(a). By $\sim$12 qubits the real device hovers around $50\%$ algorithmic error, which is indistinguishable from randomly guessing solutions to the problem from the space of all possibilities.