On the Gap of Hamiltonians for the Adiabatic Simulation of Quantum Circuits

Abstract

The time or cost of simulating a quantum circuit by adiabatic evolution is determined by the spectral gap of the Hamiltonians involved in the simulation. In "standard" constructions based on Feynman's Hamiltonian, such a gap decreases polynomially with the number of gates in the circuit, L. Because a larger gap implies a smaller cost, we study the limits of spectral gap amplification in this context. We show that, under some assumptions on the ground states and the cost of evolving with the Hamiltonians (which apply to the standard constructions), an upper bound on the gap of order 1/L follows. In addition, if the Hamiltonians satisfy a frustration-free property, the upper bound is of order 1/L 2 . Our proofs use recent results on adiabatic state transformations, spectral gap amplification, and the simulation of continuous-time quantum query algorithms. They also consider a reduction from the unstructured search problem, whose lower bound in the oracle cost translates into the upper bounds in the gaps. The impact of our results is that improving the gap beyond that of standard constructions (i.e., 1/L 2 ), if possible, is challenging.

History-state Hamiltonians are critical

“…In the quantum setting, a paper by Brandao and Harrow [7] proves (amongst many other results) that an analogous intuition is true in a different context: any quantum circuit that achieves a computational speedup over classical algorithms must at some point during the computation produce a quantum state that is "critical", in the sense that it has longrange correlations. In the context of adiabatic quantum computation, Gant and Somma [12] used query complexity bounds to prove that the spectral gap along an adiabatic path must close as O(1/n) for Feynman-Kitaev-style Hamiltonian constructions. Our results lend more support to the intuition that computational complexity is related to criticality of Hamiltonians.…”

Analysis and limitations of modified circuit-to-Hamiltonian constructions

“…The notion of modifying the weights in Feynman's circuit Hamiltonian goes back to at least 1985, when Peres [21] considered modified weights for improving the probability of measuring the output of the computation in the ballistic Hamiltonian model. In the context of universal adiabatic computation Ganti and Somma [22] derived a limitation on improving the spectral gap of circuit Hamiltonians by applying the lower bound on the Gover search problem. In [16], the authors go beyond unitary circuit include certain projective measurements {Π, 1−Π} (ones whose outcome probabilities are independent of all valid input state at that computational step).…”

“…( 2). First, the problem of upper bounding the spectral gap of universal adiabatic constructions was addressed before [22] by combining the quantum lower bound for unstructured search with the technique of spectral gap amplification. This previous work found a general Õ(T −1 ) bound (where the tilde hides logarithmic factors) on the spectral gap of any adiabatic Hamiltonian, an Õ(T −2 ) gap for any frustration-free adiabatic Hamiltonian, and finally an Õ(T −2 ) bound on the spectral gap of modified Feynman Hamiltonians of the form eq.…”

ASCR Workshop on Quantum Computing for Science

“…Nevertheless, even using small quantum codes can have a positive impact. Jordan et al developed the first AQC codes [38], and since then several others have also been developed [39,41,42,96]. Some of these have been applied to actual hardware, as depicted in Figure 2, with an obvious improvement in performance.…”