Ph.D. Thesis - Physics

Chapter 3
Quantum simulation of the BCS
Hamiltonian
Quantum simulation has great potential for calculating properties of quantum-mechanical
systems that are intractable on classical computers, as we discussed in Ch. 1. However, when
considering the power of a quantum simulator relative to a classical one, it is necessary to
also consider the precision in the final answer that can possibly be obtained, an issue we
began to explore at the close of Ch. 2. In the problem we consider here, the number
of gates needed to compute the answer is indeed polynomial in the size of the problem
Hilbert space, but is nevertheless exponential in the number of digits of precision one may
obtain. Therefore, although the quantum algorithm is in some sense “more tractable,” it
is not straightforward to determine (or guess), for a given problem, whether a classical or
quantum algorithm will actually produce a result with a given precision using a smaller
number of gates.
In addition to the question of ultimate precision, there is the fact that all implementa-
tions of quantum simulation rely on classical controls that are never perfect; these affect
the final result either by inducing decoherence or by leading to systematic errors in the
final result. Indeed, even though systematic control errors can in principle be perfectly
compensated (Ref. [BHC04]), the number of gates required to do this scales (in general)
exponentially with the residual error. This implies that even if a general quantum algorithm
may be efficient, the final precision may depend on the specific technology used to imple-
ment it. Therefore, implementations of small quantum simulations using various quantum
technologies (NMR, ion traps, etc.) are worthwhile, in that the effects of faulty controls for
each may be understood.
In this chapter, we report an experiment to test the limitations on the precision of a
three-qubit algorithm to compute the low-lying spectrum of a class of pairing Hamiltonians
[WBL02]. Our work is somewhat more ambitious than that summarized in the last chapter,
for two reasons. First, we implement a Hamiltonian that contains non-commuting parts,
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