Fault-tolerant quantum computation versus Gaussian noise

We study the robustness of a fault-tolerant quantum computer subject to Gaussian non-Markovian quantum noise, and we show that scalable quantum computation is possible if the noise power spectrum satisfies an appropriate “threshold condition.” Our condition is less sensitive to very-high-frequency noise than previously derived threshold conditions for non-Markovian noise.

Figure 1

(Color online) The four-point correlation function for a free field can be expressed in terms of products of two-point correlation functions by summing over all “contractions,” where each contraction divides the four points into two unordered pairs.

Figure 2

(Color online) The diagram in (a) represents the norm squared of the system-bath state $U_{SB}|{\Psi }_{SB}^0⟩$. By bending it into a hairpin shape we obtain the “Keldysh diagram” shown in (b). A “marked location” in the circuit appears twice in the Keldysh diagram, once on the upper branch (as a contribution to $U_{SB}$) and once on the lower branch (as a contribution to $U_{SB}^{†}$).

Figure 3

(Color online) A Keldysh diagram contributing to $⟨{\Psi }_{SB}^{\text{bad}}\left({\mathcal{I}}_{r=1}\right)|{\Psi }_{SB}^{\text{bad}}\left({\mathcal{I}}_{r=1}\right)⟩$, where black dots are insertions, lines connecting pairs of dots are contractions, and the shaded rectangle on each branch indicates the one marked location. In this diagram, the marked location is bad because there is an insertion inside the marked location on each branch.

Figure 4

(Color online) Skeleton Keldysh diagrams contributing to $⟨{\Psi }_{SB}^{\text{bad}}\left({\mathcal{I}}_{r=1}\right)|{\Psi }_{SB}^{\text{bad}}\left({\mathcal{I}}_{r=1}\right)⟩$, showing the earliest insertion of $H_{SB}$ inside the marked location on both branches. For the class 1 diagram shown in (a), the earliest insertions on the two branches are contracted with one another and for the class 2 diagram shown in (b), the earliest insertions are contracted with other insertions elsewhere. Other diagrams in each class are obtained by dressing the skeleton diagrams with additional insertions and contractions, except that no insertions are allowed inside the marked locations at times before the earliest insertion.

Figure 5

(Color online) Skeleton Keldysh diagrams contributing to $⟨{\Psi }_{SB}^{\text{bad}}\left({\mathcal{I}}_{r=2}\right)|{\Psi }_{SB}^{\text{bad}}\left({\mathcal{I}}_{r=2}\right)⟩$, showing the earliest insertion of $H_{SB}$ inside each of two marked locations on both branches. There are three skeleton diagrams with two internal contractions, six skeleton diagrams with one internal contraction, and one skeleton diagram with no internal contraction, where we say that a contraction is internal if it links two earliest insertions.

Figure 6

(Color online) The three connected Keldysh diagrams for a single qubit subject to Gaussian dephasing noise. The first two diagrams are good because ${\sigma }_z$ is inserted an even number of times on each branch and the third diagram is bad because ${\sigma }_z$ is inserted an odd number of times on each branch.