Simon's Period Finding on a Quantum Annealer

TL;DR

This work investigates Simon's period-finding algorithm in a noisy, non-gate-based setting by reformulating it as a QUBO and implementing it via adiabatic quantum computation on D-Wave quantum annealers. By introducing penalties on oracle-output transitions, the ground state becomes degenerate in a controlled way, encoding two inputs that the oracle maps to the same output. Hardware experiments show that the method is robust to noise and benefits from avoiding post-processing, but the overall runtime scales exponentially with problem size, while classical solvers maintain quadratic scaling in the tested range. The study highlights both a practical path for leveraging quantum annealing for hidden-subgroup-type problems and the current hardware limitations compared to classical approaches.

Abstract

Dating to 1994, Simon's period-finding algorithm is among the earliest and most fragile of quantum algorithms. The algorithm's fragility arises from the requirement that, to solve an n qubit problem, one must fault-tolerantly sample O(n) linearly independent values from a solution space. In this paper, we study an adiabatic implementation of Simon's algorithm that requires a constant number of successful samples regardless of problem size. We implement this algorithm on D-Wave hardware and solve problems with up to 298 qubits. We compare the runtime of classical algorithms to the D-Wave solution to analyze any potential advantage.

Figures (5)

• Figure 1: Energy spectrum for oracle of size $n=3$ (meaning there are 3 qubits in the input register, and $3n-2=7$ qubits total across all registers). The top panel gives the spectrum of the QUBO with no penalties applied. In this setting, all the valid evaluations of the oracle appear in a degenerate ground state. Hence, sampling this QUBO yields one of these values at random, which queries the oracle with a random input. The bottom panel, however, gives the spectrum with penalties $p_1=2$ and $p_2=-2$. In this case, two valid oracle evaluations that share a unique output are isolated in the ground state. Retrieving both inputs that map to this output is sufficient to solve the problem.
• Figure 2: The embedding of the QUBO \ref{['eq:penalizedGeneralQUBO']} on D-Wave Advantage 5.4 for $n=50$ (50 qubit input to the oracle, 148 total qubits). The structure of the QUBO is such that each variable interacts directly with three or six others. This connectivity pattern is sufficiently sparse to embed the problem directly on a D-Wave device without introducing swap operations or ancilla qubits.
• Figure 3: Success percentage for our adiabatic Simon's algorithm as a function of problem size. As before, the problem size ($n$) denotes the number of qubits in the input register; the total number is $3n-2$. The number of shots per problem is fixed at 4000, with an annealing time of 100$\mu s$. The experiment was executed on the D-Wave Advantage2 Prototype 2.6 and Advantage 5.4 systems. A successful run samples both target values in the degenerate ground state at least once. We find that for $n \leq 40$, the success rate is well fit by a Gaussian curve, whereas for $n > 40$, an exponential curve provides a better fit.
• Figure 4: Approximate runtime as a function of problem size for our adiabatic Simon's algorithm and a classical tree search QUBO solver implemented by D-Wave. The runtime for the annealing algorithm is estimated from the number of shots required to expect to sample from the ground state twice, yielding a $50\%$ chance of solving the problem if the degenerate pair is equally probable. The classical algorithm manifests quadratic asymptotic scaling, while our annealing algorithm scales exponentially with problem size.
• Figure 5: Runtime as a function of problem size for our adiabatic Simon's algorithm and the VeloxQ classical QUBO solver, for $n\in[2,50]$ and 1024 shots. As before, the experiment was executed on the D-Wave Advantage 5.4 and Advantage2 Prototype 2.6 systems. In the region tested, VeloxQ exhibited a slight downward trend in runtime, while the D-Wave systems showed upward trends.