Quantum error correction for quantum memories

Active quantum error correction using qubit stabilizer codes has emerged as a promising, but experimentally challenging, engineering program for building a universal quantum computer. In this review the formalism of qubit stabilizer and subsystem stabilizer codes and their possible use in protecting quantum information in a quantum memory are considered. The theory of fault tolerance and quantum error correction is reviewed, and examples of various codes and code constructions, the general quantum error-correction conditions, the noise threshold, the special role played by Clifford gates, and the route toward fault-tolerant universal quantum computation are discussed. The second part of the review is focused on providing an overview of quantum error correction using two-dimensional (topological) codes, in particular, the surface code architecture. The complexity of decoding and the notion of passive or self-correcting quantum memories are discussed. The review does not focus on a particular technology but discusses topics that will be relevant for various quantum technologies.

Figure 1

Measuring parity checks in the quantum-circuit way. The meter denotes measurement in the $\{|0⟩,|1⟩\}$ basis or $M_Z$. (a) Circuit to measure the $\pm 1$ eigenvalues of a unitary multiqubit Pauli operator $P$. The gate is the controlled-$P$ gate which applies $P$ when the control qubit is 1 and $I$ if the control qubit is 0. (b) Realizing the evolution $\exp (-i\theta P/2)$ itself [with $R_x(\theta )=\exp (-i\theta X/2)$]. (c) Realization of circuit (a) using cnot gates when $P=X_1{X}_2{X}_3{X}_4$. (d) Realization of circuit (a) using cnot gates when $P=Z_1{Z}_2{Z}_2{Z}_4$.

Figure 2

The nine-qubit [[9,1,3]] Shor code with black qubits on the vertices. The stabilizer of Shor’s code is generated by the weight-2 $Z$ checks as well as two weight-6, double-row $X$ checks ${\mathbf{X}}_{=,1}=X_1{X}_2{X}_3{X}_4{X}_5{X}_6$ and ${\mathbf{X}}_{=,2}$. An alternative way of measuring ${\mathbf{X}}_{=,1}$ and ${\mathbf{X}}_{=,2}$ is by measuring the weight-2 $X$ checks. One can similarly define two weight-6, double-column $Z$ checks ${\mathbf{Z}}_{\parallel ,1}$ and ${\mathbf{Z}}_{\parallel ,2}$ as products of elementary weight-2 $Z$ checks. See also Fig. 14.

Figure 3

Amplitude of code words for the stabilizer code with commuting checks $S_q(\alpha )=e^{2i\pi \widehat{q}/\alpha }$ and $S_p(\alpha )=e^{-2i\widehat{p}\alpha }$ which encodes a qubit in an oscillator. From [79].

Figure 4

Code concatenation: each qubit in the circuit on the left is replaced by an encoded block of qubits in the circuit on the right. The gate $G$ in the circuit $M_{r-1}$ is replaced by a rectangle consisting of the fault-tolerant encoded realization of the gate ($G_{\text{fault}\text{−}\text{tol}}$) followed by error-correcting steps ($E$). The process can be repeated for every elementary qubit and gate in the new circuit $M_r$.

Figure 5

The so-called one-bit teleportation circuits. The measurement denoted by the meter is a measurement in the $Z$ basis and determines whether to do a Pauli operation on the output qubit: for outcome $M_Z=+1$ no correction is performed. From [160].

Figure 6

Using the ancilla $T|+⟩$ in the dashed box, one can realize the $T$ gate by doing a corrective operation $SX$.

Figure 7

Surface code on an $L\times L$ lattice. On every edge of the (black) lattice there is a qubit, in total $L^2+(L-1{)}^2$ qubits (depicted is $L=8$). Two types of local parity checks $A_s$ and $B_p$ each act on four qubits, except at the boundary where they act on three qubits. The subspace of states which satisfy the parity checks is two dimensional and hence represents a qubit. $\overline{Z}$ is any $Z$ string connecting the north to the south boundary, which is referred to as “rough”, while $\overline{X}$ is any $X$ string connecting the east to west “smooth” boundary running through vertices of the dual lattice.

Figure 8

1D cross section of the lattice in space and time. Gray links correspond to nontrivial $-1$ syndromes. Errors that could have caused such a syndrome are represented by black links. Horizontal black links are qubit errors while vertical black links are parity check measurement errors. Note that a possible error $E$ has the same boundary as the gray defect links: a likely error $E$ (in the bulk) can be found by looking for a minimum-weighted matching of the end points of the gray links. From [48].

Figure 9

The Hadamard gate on a sheet which encodes a single-logical qubit as in Fig. 7; see also [88]. After performing a Hadamard gate on each qubit, the rotated code lattice is gradually rotated back to its original orientation by adding and taking away qubits and stabilizer checks at the boundaries.

Figure 10

A cnot gate via two-qubit quantum measurements. Here $M_{XX}$ measures the operator $X\otimes X$, etc. The ancilla qubit in the middle is discarded after the measurement disentangles it from the other two input qubits. Each measurement has equal probability for outcome $\pm 1$ and Pauli corrections [not shown, see Eq. (9)] depending on these measurement outcomes are done on the output target qubit.

Figure 11

Two sheets (outer) are merged at their rough boundary by placing a row of (center) ancilla qubits in the $|0⟩$ state at their boundary and measuring the parity checks of the entire sheet. For a similar smooth merge, the ancillary qubits in between the two sheets are prepared in the $|+⟩$ state; see the $INT$ and $C$ sheets in Fig. 12. From [88].

Figure 12

Using an ancilla ($INT$) qubit sheet we can perform a cnot gate between the control ($C$) and target ($T$) sheets by a sequence of mergings and splittings between the sheets. From [88].

Figure 13

(a) A smooth hole is created by removing a block of plaquette operators and the star operators acting on qubits in the interior of the block. The $Z$ loop around the hole is $\overline{Z}$ while $\overline{X}$ is an $X$ string to the boundary. The qubits inside the hole (four in the picture) are decoupled from the lattice. (b) Two smooth holes can make one smooth qubit and two rough holes can make one rough qubit so that moving a smooth hole around a rough hole realizes a cnot gate.

Figure 14

(a) [[16,1,4]] Bacon-Shor code with $\overline{X}$, a row of $X$’s, and $\overline{Z}$, a column of $Z$’s. The stabilizer generators are double columns of $Z$’s, ${\mathbf{Z}}_{\parallel ,i}$ (one is depicted) and double rows of $X$’s, ${\mathbf{X}}_{=,j}$. (b) Decoding for $X$ errors (or $Z$ errors in the orthogonal direction). Black dots denote the places where the double-column parity checks ${\mathbf{Z}}_{\parallel ,i}$ have eigenvalue $-1$ (defects). The $X$ error string $E$ has $X$ errors in the fattened region and no errors elsewhere and $E_c$ is its complement. Clearly the string $E$ has lower weight than $E_c$ and is chosen as the likely error.

Figure 15

(a) In order to measure the $XX$ and $ZZ$ operators one can place ancilla qubits (open dots) in between the data qubits. Such an ancilla qubit interacts with the two adjacent data qubits to collect the syndrome. (b) Alternatively, to measure ${\mathbf{Z}}_{\parallel ,i}$ one can prepare a three-qubit entangled cat state $(1/\sqrt{2})(|000⟩+|111⟩)$ (vertical line of dots) which interacts locally with the adjacent system qubits. ${\mathbf{X}}_{=,1}$ could be measured by preparing a cat state for the ancilla qubits placed at, say, the horizontal lines of dots. The ancilla qubits at the open dots can be used to prepare the cat states.

Figure 16

Small example of the oscillator surface code where oscillators on the edges are locally coupled with plaquette and star operators so as to define an encoded oscillator with logical, nonlocal displacements $\overline{X}(d)$ and $\overline{Z}(c)$. The realization of the four-oscillator interaction will require strong four-mode squeezing in either position (at plaquettes) or momenta (at stars).

Figure 17

Subsystem surface code on a lattice of size $L\times L$ with $L^2$ square plaquettes (depicted is $L=3$). The qubits exist on the edges and vertices of the plaquettes and are acted upon by weight-3 $X$- and $Z$-triangle operators (which are modified to become weight-2 operators at the boundary). The stabilizer checks are weight 6 except at the boundary. From [30].