Clocks in Feynman's computer and Kitaev's local Hamiltonian: Bias, gaps, idling, and pulse tuning

Abstract

We present a collection of results about the clock in Feynman's computer construction and Kitaev's Local Hamiltonian problem. First, by analyzing the spectra of quantum walks on a line with varying endpoint terms, we find a better lower bound on the gap of the Feynman Hamiltonian, which translates into a less strict promise gap requirement for the QMA-complete Local Hamiltonian problem. We also translate this result into the language of adiabatic quantum computation. Second, introducing an idling clock construction with a large state space but fast Cesaro mixing, we provide a way for achieving an arbitrarily high success probability of computation with Feynman's computer with only a logarithmic increase in the number of clock qubits. Finally, we tune and thus improve the costs (locality, gap scaling) of implementing a (pulse) clock with a single excitation.

Analysis and limitations of modified circuit-to-Hamiltonian constructions

“…More recently, Caha, Landau, and Nagaj have presented a compilation of several results on modified circuit Hamiltonians [17]. Most of these results are distinct from results in this work, with the exception of theorem 7 which was discovered by us independently around the same time [19].…”

“…We capture this in the following theorem. Note that this result has been independently proven using Jordan's lemma type techniques by Caha, Landau, and Nagaj [17] and us [19]. In this work, we include a direct proof not based on Jordan's lemma, which can be found in appendix A.2.…”

“…Based on ideas by Feynman [9] and cast into its current form by Kitaev [1], the construction remains relatively little changed, having undergone only some gradual evolutions since its modern introduction [6,[10][11][12][13][14][15]. Only recently steps have been undertaken to analyse modifications in depth [16], and revisit the spectral properties of Kitaev's original construction [17].…”

Good approximate quantum LDPC codes from spacetime circuit Hamiltonians

“…The register C, called the clock register, indicates how many gates have been applied to the all zeroes state, which is stored in register S (called the state register) containing an initial state |ψ and ancillas. Although this state has only a 1/(T + 1) fidelity with the output of the circuit, the standard technique for increasing the overlap to be inverse polynomially close to 1 is to pad the end of the circuit with identity gates (for recent work on more efficient methods for biasing the history state towards its endpoints, see [BC18,CLN18]). This technique allows history states to capture approximate versions of QECC that have efficient encoding circuits.…”

How Fast Do Quantum Walks Mix?

“…However, given a network of n nodes, one can define the limiting distribution of quantum walk on the network as the long-time average probability distribution of finding the walker in each node [10]. Of particular interest is the quantum mixing time: starting from some initial state, the minimum time after which the underlying quantum walk remains close to its limiting distribution.The importance of the problem of mixing for quantum walks cannot be overstated: this is at the heart of quantum speedups for a number of quantum algorithms [2,11] and is also key to demonstrating the equivalence between the standard (circuit) and Hamiltonian-based models of quantum computation [12,13]. Unfortunately, no general result exists for quantum mixing time on networks: it has been estimated for a handful of specific graphs (graphs and networks are used interchangeably throughout the letter) such as hypercubes, d-dimensional lattices etc., and is known to be slower than its classical counterpart for some graphs while faster in the case of others [10,[14][15][16][17][18][19][20][21].…”