Ph.D. Thesis - Physics

Renormalization Group (DMRG), with which, for example, a pairing model with N ≈ 400
has been solved to five digits of precision [DS99]. However, DMRG is still a fundamentally
approximate method. In this chapter we explore an alternative quantum algorithm that is,
in principle, numerically exact.
3.2 The Wu-Byrd-Lidar proposal
In 2002, Wu, Byrd, and Lidar (WBL) proposed a method for simulating the BCS Hamilto-
nian on a NMR-type quantum computer [WBL02]. In this section we will, following their
paper, explain how to use a digital NMR-type quantum simulator to calculate the low-lying
spectrum of the pairing Hamiltonian. We are concerned with finding the energy gap ∆
between the ground and excited states for a specified Hamiltonian. Wu et al. prove that ∆
can be computed using O(N 4 ) steps, where N is the number of qubits, and also the number
of quantum states included in the simulation, equivalent to the number of modes N above.
They begin by mapping the Fermionic creation and annihilation operators to qubit
operators. A computational |1〉 state signifies the existence of a Cooper pair in one of
the possible modes, while |0〉 is the vacuum state for that mode. Thus, the total number
of qubits in the |1〉 state signifies the number of Cooper pairs, and the total number of
qubits translates into the total number of modes that might be occupied. Accordingly, a
measurement of the number operator nm = (Zm + 1)/2 yields 1 if the mode m is excited,
and 0 otherwise. The operator σ + m (σ − m) signifies the creation (annihilation) of a Cooper
pair in mode m: σ + m ↦→ c−mcm and σ − m ↦→ c † mc †
−m
The above identification holds for pairing Hamiltonians that simulate Fermionic particle-
particle interactions; thus, it is appropriate for the BCS Hamiltonian. Two other cases are
given in their paper, but for simplicity, we will restrict our discussion to the case that we
actually implemented experimentally (the BCS Hamiltonian). From the above point, it is
simple to write HBCS in terms of qubit operators, using the identities σ + = (X +iY )/2 and
σ − = (X − iY )/2. The final Hamiltonian that we use in our NMR implementation is
HBCS =
n
m=1
νm
2 (−Zm) +
m