26 to fast-forward low-lying states in a quantum system. A subspace-search variational eigensolver was employed in ref. This circuit was used to simulate the Ising model on Cloud QCs 31. 30 exploited the exact solvability of the transverse Ising model to formulate a quantum circuit for its exact diagonalization, allowing for fast-forwarding. For example, commuting local Hamiltonians 27, quadratic fermionic Hamiltonians 27, and continuous-time quantum walks on particular graphs 29 can all be fast-forwarded. Hamiltonians that allow fast-forwarding are precisely those that lead to violations of time–energy uncertainty relations and equivalently allow for precise energy measurements 27. However, there are particular Hamiltonians that can be fast-forwarded, which means that the quantum circuit depth does not need to grow significantly with simulation time. This result is known as the “no fast-forwarding theorem”, and holds both for a typical unknown Hamiltonian 27 and for the query model setting 28. Simulating the dynamics of a quantum system for time T typically requires Ω( T) gates so that a generic Hamiltonian evolution cannot be achieved in sublinear time.
Both approaches have the potential to outperform Suzuki–Trotter-based methods in the NISQ era. Of the variational dynamical simulation methods, some are based on knowledge of low-lying excited states 26, and some are iterative in time 23, 24, 25. In addition, some variational algorithms simulate system dynamics 23, 24, 25, 26. Variational algorithms have been introduced for finding ground and excited states 15, 16, 17, 18, and for other applications 19, 20, 21, 22. In the current noisy intermediate-scale quantum (NISQ) era, variational quantum simulation (VQS) methods are expected to be important. Quantum emulation and Suzuki–Trotter-based QSs have seen proof-of-principle demonstrations 2, 3, 4, 5, 6, 14, while Taylor expansion-based QSs have the best asymptotic scaling and will likely have application for fault-tolerant quantum computers (QCs) of the future. Current approaches include quantum emulation (or analogue QS) 2, 3, 4, 5, 6, Suzuki–Trotter-based methods 7, 8, 9, 10, and Taylor expansion-based QSs using linear combinations of unitaries 11, 12, 13. Accelerated QS would impact fields, including chemistry, materials science, and nuclear and high-energy physics. Quantum simulation (QS) was the earliest proposed example of a quantum algorithm that could outcompete the best classical algorithm 1.
#Quantumwise diagonalization error how to
Finally, we show how to estimate energy eigenvalues using VFF. In addition, we implement VFF on Rigetti’s QC to demonstrate simulation beyond the coherence time. We implement VFF for the Hubbard, Ising, and Heisenberg models on a simulator. Our error analysis provides two results: (1) the simulation error of VFF scales at worst linearly in the fast-forwarded simulation time, and (2) our cost function’s operational meaning as an upper bound on average-case simulation error provides a natural termination condition for VFF. VFF seeks an approximate diagonalization of a short-time simulation to enable longer-time simulations using a constant number of gates. Here, we present a hybrid quantum-classical algorithm, called variational fast forwarding (VFF), for decreasing the quantum circuit depth of QSs. Trotterization-based, iterative approaches to quantum simulation (QS) are restricted to simulation times less than the coherence time of the quantum computer (QC), which limits their utility in the near term.