Robustness of adiabatic quantum computation

The method of quantum computation by adiabatic evolution has been proposed as a general way of solving combinatorial search problems on a quantum computer [1]. Whereas a conventional quantum algorithm is implemented as a sequence of discrete unitary transformations that form a quantum circuit involving many energy levels of the computer, the adiabatic algorithm works by keeping the state of the quantum computer close to the instantaneous ground state of a Hamiltonian that varies continuously in time. Therefore, an imperfect quantum computer implementing a conventional quantum algorithm might experience different sorts of errors than an imperfect adiabatic quantum computer. In fact, we claim that an adiabatic quantum computer has an inherent robustness against errors that might enhance the usefulness of the adiabatic approach. The adiabatic algorithm works by applying a timedependent Hamiltonian that interpolates smoothly from an initial Hamiltonian whose ground state is easily prepared to a final Hamiltonian whose ground state encodes the solution to the problem. If the Hamiltonian varies sufficiently slowly, then the quantum adiabatic theorem guarantees that the final state of the quantum computer will be close to the ground state of the final Hamiltonian, so a measurement of the final state will yield a solution of the problem with high probability. This method will surely succeed if the Hamiltonian changes slowly. But how slow is slow enough? Unfortunately, this question has proved difficult to analyze in general. Some numerical evidence suggests the possibility that the adiabatic algorithm might efficiently solve computationally interesting instances of hard combinatorial search problems, outperforming classical methods [1–4]. Whether the adiabatic algorithm provides a definite speedup over classical methods remains an interesting open question. As we will discuss in Sec. II, the time required by the algorithm for a particular instance can be related to the minimum gap ∆ between the instantaneous ground state and the rest of the spectrum. Roughly speaking, the required time goes like ∆−2. Thus, if ∆−2 increases only polynomially with the size of the problem, then so does the time required to run the algorithm. However, determining ∆ has not been possible in general. Our objective in this paper is not to explore the computational power of the adiabatic model, but rather to investigate its intrinsic fault tolerance. Since quantum computers are far more susceptible to making errors than classical digital computers, fault tolerant protocols will be necessary for the operation of large-scale quantum computers. General procedures have been developed that allow any quantum algorithm to be implemented fault tolerantly on a universal quantum computer [5], but these involve a substantial computational overhead. Therefore, it would be highly advantageous to weave fault tolerance into the design of our quantum hardware. We therefore will regard adiabatic quantum computation not as a convenient language for describing a class of quantum circuits, but as a proposed physical implementation of quantum information processing. We do not cast the algorithm into the conventional quantum computing paradigm by approximating it as a sequence of discrete unitary transformations acting on a few qubits at a time. Instead, suppose we can design a physical device that implements the required time-dependent Hamiltonian with reasonable accuracy. We then imagine implementing the algorithm by slowly changing the parameters that control the physical Hamiltonian. How well does such a quantum computer resist decoherence, and how well does it perform if the algorithm is imperfectly implemented? Regarding resistance to decoherence, we can make a few simple observations. The phase of the ground state has no effect on the efficacy of the algorithm, and therefore dephasing in the energy eigenstate basis is presumably harmless. Only the interactions with the environ-