Adiabatic Quantum Computing relies on the quantum adiabatic theorem, which states that a quantum system evolves along its ground state with time if the governing Hamiltonian varies infinitely slowly. However, practical limitations force computations to be performed within limited times, exposing the system to transitions into excited states, and thereby reducing the success probability. Here we investigate the counterintuitive hypothesis that incorporating stochastic noise, specifically noise driven by fractional Brownian motion, in a non-Markovian setup can enhance the performance of adiabatic quantum computing by improving its success probability at limited evolution times. The study begins by developing the mathematical framework to introduce stochastic noise multiplicatively into the Schrödinger equation, resulting in a stochastic Schrödinger equation. To preserve Itô integrability within the non-Markovian framework, a semimartingale approximation for fractional Brownian motion is employed. We perform numerical simulations to compare the performance of the quantum adiabatic algorithm with and without noise driven by fractional Brownian motion using the NP-complete Exact Cover-3 problem, transformed into the Ising model. Our results exhibit an improvement in success probability in the presence of noise driven by fractional Brownian motion with Hurst parameter $0
Stochastic processes play a fundamental role in physics, mathematics, engineering and finance. One potential application of quantum computation is to better approximate properties of stochastic processes. For example, quantum algorithms for Monte Carlo estimation combine a quantum simulation of a stochastic process with amplitude estimation to improve mean estimation. In this work we study quantum algorithms for simulating stochastic processes which are compatible with Monte Carlo methods. We introduce a new ``analog'' quantum representation of stochastic processes, in which the value of the process at time t is stored in the amplitude of the quantum state, enabling an exponentially efficient encoding of process trajectories. We show that this representation allows for highly efficient quantum algorithms for simulating certain stochastic processes, using spectral properties of these processes combined with the quantum Fourier transform. In particular, we show that we can simulate
T timesteps of fractional Brownian motion using a quantum circuit with gate complexity
polylog(T), which coherently prepares the superposition over Brownian paths. We then show this can be combined with quantum mean estimation to create end to end algorithms for estimating certain time averages over processes in time
O(polylog(T)ε−c) where $3/2
Quantum computation has revolutionary potential for speeding algorithms and for simulating quantum systems such as molecules. We report here a quantum computer design that performs universal quantum computation within a single non-degenerate ground state protected from decohering noise by an energy gap that we argue is system-size-independent. Closely analogous to a traditional electric circuit, it substantially changes the requirements for quantum computer construction, easing measurement, timing, and heating problems. Using the standard adiabatic condition, we present evidence that this design permits "quantum concurrent processing" distributing a quantum computation among extra qubits to perform a quantum algorithm of N gates in an amount of time that scales with the square root of N. One consequence of our work is a fixed gap version of adiabatic quantum computation, which several arguments hinted could be impossible.