Craig Gidney

When designing quantum circuits for a given unitary, it can be much cheaper to achieve a good approximation on most inputs than on all inputs. In this work we formalize this idea, and propose that such "optimistic quantum circuits" are often sufficient in the context of larger quantum algorithms. For the rare algorithm in which a subroutine needs to be a good approximation on all inputs, we provide a reduction which transforms optimistic circuits into general ones. Applying these ideas, we build an optimistic circuit for the in-place quantum Fourier transform (QFT). Our circuit has depth O(log(n/ϵ)) for tunable error parameter ϵ, uses n total qubits, i.e. no ancillas, is local for input qubits arranged in 1D, and is measurement-free. The circuit's error is bounded by ϵ on all input states except an ϵ-sized fraction of the Hilbert space. The circuit is also rather simple and thus may be practically useful. Combined with recent QFT-based fast arithmetic constructions, the optimistic QFT yields factoring circuits of nearly linear depth using only 2n + O(n/log n) total qubits. Additionally, we apply our reduction technique to yield an approximate QFT with well-controlled error on all inputs; it is the first to achieve the asymptotically optimal depth of O(log (n/ϵ)) with a sublinear number of ancilla qubits. The reduction uses long-range gates but no measurements.

We prove by construction that the Bravyi-Poulin-Terhal bound on the spatial density of stabilizer codes does not generalize to stabilizer circuits. To do so, we construct a fault tolerant quantum computer with a coding rate above 5$\%$ and quasi-polylog time overhead, out of a line of qubits with nearest-neighbor connectivity, and prove it has a threshold. The construction is based on modifications to the tower of Hamming codes of Yamasaki and Koashi (Nature Physics, 2024), with operators measured using a variant of Shor’s measurement gadget.

Abstract We show how to achieve the highest efficiency yet for simulations with arbitrary basis sets by using a representation of the Coulomb operator known as tensor hypercontraction (THC). We use THC to express the Coulomb operator in a non-orthogonal basis, which we are able to block encode by separately rotating each term with angles that are obtained via QROM. Our algorithm has the best complexity scaling for an arbitrary basis, as well as the best complexity for the specific case of FeMoCo. By optimising the surface code resources, we show that FeMoCo can be simulated using about 4 million physical qubits and 3.5 days of runtime, assuming 1 s cycle times and physical gate error rates no worse than 0.1%.

Abstract We compile explicit circuits and evaluate the computational cost for heuristic-based quantum algorithms for combinatorial optimization. We consider several variants of quantum-accelerated simulated annealing as well as adiabatic algorithms, quantum-enhanced population transfer, the quantum approximate optimization algorithm, and other approaches. We provide novel methods for executing the bottleneck subroutines for these heuristics, and our methods can easily be applied to other algorithms where numerical performance matters. We estimate how quickly the subroutines could be executed on a modestly sized superconducting-qubit-based quantum computer with surface code error correction. We conclude that quadratic speedups for heuristic-based quantum optimization algorithms are insufficient for early quantum computers to beat present day classical computers.