Faster ground state preparation and high-precision ground energy estimation with fewer qubits

Abstract

We propose a general-purpose quantum algorithm for preparing ground states of quantum Hamiltonians from a given trial state. The algorithm is based on techniques recently developed in the context of solving the quantum linear systems problem [1]. We show that, compared to algorithms based on phase estimation, the runtime of our algorithm is exponentially better as a function of the allowed error, and at least quadratically better as a function of the overlap with the trial state. We also show that our algorithm requires fewer ancilla qubits than existing algorithms, making it attractive for early applications of small quantum computers. Additionally, it can be used to determine an unknown ground energy faster than with phase estimation if a very high precision is required.

A hybrid algorithm framework for small quantum computers with application to finding Hamiltonian cycles

“…However, the rate at which the number of qubits we can manipulate with relevant precision and coherence times currently grows provides significant motivation for studying potential uses of size-limited quantum computers. Complementary to research dedicated to solving smallyet-hard simulation and ground-state problems [21][22][23], which are promising applications for really small quantum computers, in this work, we investigated ways to achieve speedups of classical algorithms by exploiting quantum computers significantly smaller than the problem size. Concretely, we provide a framework for designing hybrid quantum-classical algorithms, which can allow for polynomial asymptotic speedups given a quantum computer which is any constant fraction of the problem size.…”

Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems

“…From the analysis in Ref. [30], this approach requires O(1/(γ 2 ∆ )) times of queries for U sim , where is the target accuracy (the complexity is the same up to logarithmic factors if we use the block-encoding U H instead of its time-evolution as an oracle); the number of queries to the circuit U x 0 that prepares the initial trial state is O(1/γ); and the number of extra ancilla qubits is O(log(1/( ∆)). This is non-optimal with respect to both γ and .…”

“…Several variants of phase estimation are developed to achieve better dependence on the parameters γ and [30,48,49]. The filtering method developed by Poulin and Wocjan [48] (for a task related to eigenstate filtering) improves the query complexities of U sim and U x 0 with respect to γ from O(1/γ 2 ) to O(1/γ).…”

“…The filtering method developed by Poulin and Wocjan [48] (for a task related to eigenstate filtering) improves the query complexities of U sim and U x 0 with respect to γ from O(1/γ 2 ) to O(1/γ). Ge et al [30,Appendix C] shows that the method by Poulin and Wocjan can be adapted to the ground state preparation problem so that the query complexity of U sim becomes O(1/(γ∆) log(1/ )), while the complexity of U x 0 remains O(1/γ). The number of extra ancilla qubits is O(log(1/( ∆)).…”

“…Ge et al [30] also proposed two eigenstate filtering algorithms using linear combination of unitaries (LCU) [11,21], which uses the Fourier basis and the Chebyshev polynomial basis, respectively. For both methods, the query complexities for U H and U x 0 are O(1/(γ∆) log(1/ )) and O(1/γ) respectively, and the number of extra ancilla qubits can be reduced to O(log log(1/ ) + log(1/∆)).…”