26 2. ADIABATIC QUANTUM COMPUTATION
2.4.1 AQC AND RELATED MODELS
Where does AQC fit into the complexity classes of Figure 2.3? e main result in this area is
a proof by Aharonov et al. [1] that a universal version of AQC can simulate any computation
in the quantum gate model (QGC) with at most polynomial slowdown. Combined with the
result by Farhi et al. [36] that the quantum gate model can simulate any AQC computation, this
implies that AQC is polynomially equivalent to QGC. erefore AQC—which contains the set
of computable optimization problems—must also contain complexity classes analogous to QMA
and BQP, that contain NPO and FPTAS respectively.
Van Dam et al. [105] have shown that an AQC algorithm for Grovers problem—searching
in an unordered set—exhibits quadratic speedup over classical algorithms, as is also the case with
the gate model.
Locality. e Aharonov et al. [1] proof assumes a computational model in which the initial and
final Hamiltonians H
I
and H
F
are k-local. A k-local Hamiltonian can be written as the sum of
independent Hamiltonians that each work on at most k qubits. For example, the EC algorithm
(2.31) is 2-local because H
F
is the sum of operators involving at most two qubits.
Local Hamiltonians are considered better candidates for realization on a physical quan-
tum computer which may have limited connectivity. Sometimes locality is defined according to
topology: for example, the final Hamiltonian may be restricted to involve only interactions among
neighboring qubits in a grid.
Aharonov et al. [1] show that any quantum circuit C
f
using at most L gates can be simu-
lated by an adabatic quantum computer using one of these models:
Five-local Hamiltonians, with at most O.L
5
/ expansion in problem size.
ree-local Hamiltonians, with at most O.L
14
/ problem overhead.
Two-local Hamiltonians interacting on a 2D grid of qubit cells, assuming each cell can
represent six different states.
Restrictions on the universal model. Published AQC algorithms use final Hamiltonians with
real-valued diagonal elements that exactly match the objective functions f .x/ and with off-
diagonals equal to zero. is category of Hamiltonian H
F
creates a classical ground state (with
no superposition) at the time the solution is read. A much wider choice of final Hamiltonians is
available in the general computational model; indeed Aharonov et al. [1] remark that their proof
of equivalence with the gate model assumes that the final Hamiltonian is not restricted to be
classical in this way.
Bravyi et al. [18] define a stoquastic Hamiltonian as one for which off-diagonal elements
are real and non-positive. ey show that the problem of minimizing local stoquastic Hamilto-
nians is hard for the class AM (similar to MA, a probabilistic version of NP with two rounds of
communication between the prover and the verifier).
2.4. COMPLEXITY CLASSES 27
Biamonte and Love [13] consider restrictions to the universal AQC model with an eye
toward realizability in physical machines. ey describe two categories of Hamiltonians that are
sufficient to achieve QMA-complete computation: this means the models can solve NP-hard
problems. ey also describe how to modify these models to achieve universal AQC computation.
e quantum annealing chips built by D-Wave require problem Hamiltonians specified
with reals on the diagonal and zeros on the off-diagonal, and have a connectivity structure that
is 2-local. erefore the proof of Aharonov et al. [1] of AQC equivalence with the quantum
gate model does not apply to these chips in their current design. e approximation problem
implemented in the D-Wave hardware topology has a classical PTAS; see Section 4.1 for more.

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