Fault-tolerant quantum computation against biased noise

We formulate a scheme for fault-tolerant quantum computation that works effectively against highly biased noise, where dephasing is far stronger than all other types of noise. In our scheme, the fundamental operations performed by the quantum computer are single-qubit preparations, single-qubit measurements, and conditional-phase (CPHASE) gates, where the noise in the CPHASE gates is biased. We show that the accuracy threshold for quantum computation can be improved by exploiting this noise asymmetry; e.g., if dephasing dominates all other types of noise in the CPHASE gates by four orders of magnitude, we find a rigorous lower bound on the accuracy threshold higher by a factor of 5 than for the case of unbiased noise.

Figure 1

(Color online) Scheme for achieving fault-tolerant universal quantum computation against highly biased noise.

Figure 2

(Color online) Gadget that measures ${\sigma }_z^L$ (here, $n=3$). The measurement is repeated $r$ times to ensure fault tolerance (here, $r=3$), with the repetitions staggered as shown so that the data qubits are never idle in between consecutive gates.

Figure 3

(Color online) On the left, a “one-bit teleportation” circuit. On the right, the error correction gadget.

Figure 4

(Color online) Fault-tolerant CNOT gadget. Pauli operators that update the Pauli frame (not shown) are determined by the measurement outcomes.

Figure 5

(Color online) Upper bounds on the effective noise strength ${\epsilon }^{\left(1\right)}$ for ${\mathcal{C}}_1$-protected ${\mathcal{G}}_{\mathrm{CSS}}$ operations as a function of $\epsilon$ and the bias $\epsilon ∕{\epsilon }^{\prime }$ (where for each value of $\epsilon$, we optimize over $n$ and $r$). The straight line with slope unity serves as a guide to the eye.

Figure 6

(Color online) The full circuit of the CNOT gadget (here, for compactness we have chosen $r_1=r_2=n=2$, but the structure of the circuit is similar for odd values).

Figure 7

(Color online) A CNOT gadget preceded by CNOT gadgets acting on each of its input blocks.

Figure 8

(Color online) Teleportation circuits for the Clifford group gates $Q=e^{i(\pi ∕4){\sigma }_x}$ and $S=e^{-i(\pi ∕4){\sigma }_z}$ and for the non-Clifford gate $T=e^{-i(\pi ∕8){\sigma }_z}$. For the Clifford group gates the measurement determines a Pauli operator that updates the Pauli frame, and for the non-Clifford gate the measurement determines a non-Pauli correction that must be applied in the next step.

Figure 9

(Color online) Injection of the state $\mid \psi ⟩$ into the ${\mathcal{C}}_1⊳{\mathcal{C}}_2$ block by using teleportation. After the encoded Bell state $\mid {\overline{\Phi }}_0⟩$ is prepared, one ${\mathcal{C}}_1⊳{\mathcal{C}}_2$ block is decoded to ${\mathcal{C}}_1$. Then, a Bell measurement is performed on the ${\mathcal{C}}_1$ block and the input state $\mid \psi ⟩$.

Figure 10

(Color online) A Bell state ${\mid {\Phi }_0⟩}_L$ is prepared by starting with two blocks in the state ${\mid +⟩}_L$ and then measuring ${\sigma }_z^L{\sigma }_z^L$.

Figure 11

(Color online) On the top, the CNOT gates and measurements used to implement the teleportations of ${\mathcal{C}}_2$ blocks and subblocks in Fig. 3 in [9]. On the bottom, a ${\mathcal{C}}_1$-protected implementation of the operations on the top. The outcome of a measurement of ${\sigma }_z^L{\sigma }_z^L$ or ${\sigma }_z^L{\sigma }_z^L{\sigma }_z^L$ on the bottom corresponds to the outcome of the measurement of ${\sigma }_z$ on the top with the same superscript, while the outcomes of the measurements of ${\sigma }_x^L$ on the bottom are combined as shown to give the outcomes of the measurements of ${\sigma }_x$ on the top.