Quantum computation via measurements on the low-temperature state of a many-body system

We consider measurement-based quantum computation using the state of a spin-lattice system in equilibrium with a thermal bath and free to evolve under its own Hamiltonian. Any single qubit measurements disturb the system from equilibrium and, with adaptive measurements performed at a finite rate, the resulting dynamics reduces the fidelity of the computation. We show that it is possible to describe the loss in fidelity by a single quantum operation on the encoded quantum state that is independent of the measurement history. To achieve this simple description, we choose a particular form of spin-boson coupling to describe the interaction with the environment, and perform measurements periodically at a natural rate determined by the energy gap of the system. We found that an optimal cooling exists, which is a trade-off between keeping the system cool enough that the resource state remains close to the ground state, but also isolated enough that the cooling does not strongly interfere with the dynamics of the computation. For a sufficiently low temperature we obtain a fault-tolerant threshold for the couplings to the environment.

Figure 1

Contour plot of fidelity as a function of coupling $\alpha$, to a zero temperature bath and measurement time $t$. We set $\Delta =1$ and show ten equally spaced contours between $F=0$ $\left(\alpha =0,t=\pi \right)$ and $F=1$ $\left(\alpha =0,t=2\pi n\right)$. Each shaded band corresponds to an interval of 10%, for example, the white regions correspond to a fidelity of $90\le F\le 100\%$, centered around multiples of $\tau$, the uppermost large gray region corresponds to $40\le F\le 50\%$ while the black regions correspond to $0\le F\le 10\%$.

Figure 2

Fidelity as a function of the cooling constant ${\alpha }_{\text{bath}}$ for couplings to the background (from the bottom to the top) $\gamma ={10}^{-2},{10}^{-3},{10}^{-4},\text{and}{10}^{-5},$. For each there is an optimal cooling rate that maximizes the fidelity.

Figure 3

(Color online) Schematic diagram for the localization of the logical state to the qubits ${\mathcal{Q}}_N$ after $N-1$ timesteps. The qubits to the right of ${\mathcal{Q}}_N$ are disentangled with a unitary $V_N$, while those to the left have already been measured. The qubits ${\mathcal{Q}}_{N-1}$ are in a pure product state, having just been measured.