Universal fault-tolerant measurement-based quantum computation

Certain physical systems that one might consider for fault-tolerant quantum computing where qubits do not readily interact, for instance photons, are better suited for measurement-based quantum-computational protocols. Here we propose a measurement-based model for universal quantum computation that simulates the braiding and fusion of Majorana modes. To derive our model we develop a general framework that maps any scheme of fault-tolerant quantum computation with stabilizer codes into the measurement-based picture. As such, our framework gives an explicit way of producing fault-tolerant models of universal quantum computation with linear optics using any protocol developed using the stabilizer formalism. Given the remarkable fault-tolerant properties that Majorana modes promise, the main example we present offers a robust and resource-efficient proposal for photonic quantum computation.

Figure 1

Three resource states shown in red, blue and green are composed to map an input state onto the output state under a mapping that is determined by the different resource states. A temporal direction can be assigned such that the input system is input at the initial time and the output system emerges at the final instance.

Figure 2

The one-dimensional cluster state. Physical qubits are shown as vertices ordered along a line where the left-most vertex represents the first qubit of the system and edges indicated pairs of vertices that are entangled via a controlled-phase gate. (a) A stabilizer operator denoted $C\left[\mu \right]$ shown at an arbitrary point along the lattice. Panels (b) and (c) show, respectively, representatives of the logical Pauli-X and Pauli-Z operators once the cluster state is initialized.

Figure 3

The logical operators of the one-dimensional cluster state after and before the first qubit is measured in the Pauli-X basis. In panel (a) we show the logical Pauli-X operator and in panel (b) we show the logical Pauli-Z operator after the first qubit is measured, and the measurement outcome $x_1$ is returned. Importantly, after the measurement the logical Pauli-X operator is a single-qubit Pauli-Z operator on the second qubit, whereas the logical Pauli-Z operator is supported on two physical qubits. In contrast, before the measurement the logical Pauli-Z operator is supported on a single qubit, where as the logical Pauli-X operator is a weight-two operator. We show the logical Pauli-X and logical Pauli-Z operator before the measurement in panels (c) and (d), respectively, for comparison. These are the same operators as those shown in Figs. 2 and 2, respectively.

Figure 4

(a) Logical Pauli-X and (b) logical Pauli-Z operators at time interval $t$. Each time interval contains one black vertex and one red vertex. Local to each time interval we can adjust our single-qubit measurement pattern by measuring the appropriate qubit in the Pauli-Z operator to measure either the logical Pauli-X or Pauli-Z operator given that the preceding qubits of the system are measured in the Pauli-X basis.

Figure 5

The logical Pauli-Y operator on a type-I foliated qubit where the blue ancilla qubit is coupled both to qubits $Z\left(t\right)$ and $X\left(t\right)$ of the chain. The logical operator commutes with the measurement pattern where all of the qubits along the chain are measured in the Pauli-X basis, and the ancilla qubit is measured in the Pauli-Y basis, thus allowing us to recover Pauli-Y data from the foliated qubit.

Figure 6

Two type-I foliated qubits where ancillas are coupled to perform $X_1{X}_2$ parity measurements. To the left of the figure we show a logical operator $\sim X_1{X}_2$ by coupling an ancilla, shown in blue, to qubits $X_1\left(1\right)$ and $X_2\left(1\right)$. To the right of the figure we show an element of the measurement pattern that is generated by two parity measurements that are made at different time intervals. This term is also a stabilizer of $\mathcal{F}$.

Figure 7

Two type-I foliated qubits where we use the ancilla to measure the parity of the two foliated qubits in the Pauli-Y basis. The ancilla is coupled to the physical qubits indexed $X_j\left(t\right)$ and $Z_j\left(t\right)$ for each foliated qubit $j=1,2$. We measure the ancilla in the Pauli-X basis, since there are an even number of Pauli-Y terms in the parity measurement.

Figure 8

Sketch of the foliation process and construction of the system. Time moves from left to right across the page in each figure. (a) The initial system, $\mathcal{I}$, consists of an input code, ${\mathcal{G}}_{\text{in}}$, shown with green qubits to the left of the figure, and other qubits, shown in black and red in the figure, prepared in the $|+\rangle$ state. (b) The channel system $\mathcal{K}$ is produced by applying unitary $U^C$ to $\mathcal{I}$. The edges in the figure represent controlled-phase gates prepared between pairs of qubits. Effectively, we have concatenated each of the qubits of ${\mathcal{G}}_{\text{in}}$ into the code space of a one-dimensional cluster state. (c) We produce $\mathcal{R}$ by entangling the qubits of the ancilla system, $\mathcal{A}$, to the channel system. The coupling is specified by the choice of ${\mathcal{G}}_{\text{ch.}}$. This choice determines the check operators of the foliated system, and the deformation on the input qubit. (d) The qubits of $\mathcal{R}$ are measured according to $\mathcal{M}$. This propagates the input system onto the output system, ${\mathcal{G}}_{\text{out}}$, shown in purple to the right of the figure. The measurement pattern determines the value of the stabilizer checks, and in turn the correction that must be applied to the output system.

Figure 9

The twisted surface code model. Examples of star and plaquette operators are shown explicitly on thick outlined plaquttes on blue and yellow faces, respectively. Logical operators are supported on the qubits covered by the yellow and blue dotted lines. A defect line runs from the right-hand side of the lattice to its center. An example of a modified stabilizer that is supported on the defect line is shown at (a). The green line terminates at the center of the lattice. We write the stabilizer where the defect line terminates below the lattice at (b). We focus on foliating the stabilizers that lie on the defect line, as such we number them explicitly with indices shown in black circles on the figure.

Figure 10

A foliation pattern for the defect line of the twisted surface code. The code is initialized at the bottom of the figure where the lattice that describes the stabilizers of the code are shown. Qubits that are coupled with a controlled-phase gate are connected by an edge. Qubits indexed $Z_j\left(t\right)(X_j\left(t\right)$) are colored black (red), and ancilla qubits are colored in blue (green), if they are measured in the Pauli-X (Pauli-Y) basis. Example elements $S_{\text{bulk}}$ (as in Theorem t5) which belong to $\mathcal{S}=\mathcal{C}\left(\mathcal{F}\right)\cap \mathcal{R}\cap \mathcal{M}$ are also shown.

Figure 11

Resource state where we change the time interval of some of the targets of certain ancilla qubits as is described in the main text. Qubits indexed $X_j\left(t\right)$, $Y_j\left(t\right)$, and $Z_j\left(t\right)$ are colored red, green, and black, respectively, and ancilla qubits are shown in blue. We also show the edges of the type-II foliated qubit in blue. No physical qubit has more than four incident edges. Two elements of $\mathcal{S}=\mathcal{C}\left(\mathcal{F}\right)\cap \mathcal{R}\cap \mathcal{M}$ are shown.

Figure 12

The foliated planar code. The code is extended through spacetime where the rough and smooth boundaries are shown in yellow and blue, and the output is shown by the light red face. At the interface between the rough and smooth boundaries we see the world lines of Majorana modes, depicted by thick red lines. The four modes that encode the qubit move directly along the temporal axis, and as such do not nontrivially manipulate the encoded state over the channel.

Figure 13

The input to encode an arbitrary state. (a) The initial product state used to initialize the surface code from Ref. [95]. Qubits lie on the vertices of the graph, and star(plaquette) operators lie on the blue(yellow) faces of the lattice. The system is initialized in a product state where all the qubits within the dark green(blue) boxes begin in the $+1$ eigenvalue eigenstate of the Pauli-X (Pauli-Z) operator, and the central red qubit is prepared in an arbitrary state. Measuring the stabilizers shown on the lattice initialize the code in the state of the central qubit. (b) The graph state showing initialization where the initial state shown in (a) lies at the front of the figure, and the time axis of the foliated system extends into the page. Qubits initialized in an eigenstate of the Pauli-Z operator are not drawn, as these qubits do not entangle with the other qubits of the resource state. Examples of check operators [elements of $\mathcal{C}\left(\mathcal{F}\right)\cap \mathcal{R}\cap \mathcal{M}$] are shown on the graph. (c) Initialization with the fault-tolerant cluster state shown macroscopically. Two pairs of Majorana modes are created at the site where the encoded state is initialized and their worldlines extend to the outer edges of the lattice in between the different boundaries which are determined by how the qubits of the initial system are prepared.

Figure 14

Macroscopic picture showing the fault-tolerant initialization of eigenstates of the logical Pauli operators. The time direction runs into the page. Initialization of the logical Pauli-Z (Pauli-X) operator is shown on the left-hand side (right-hand side) of the figure. The input state is prepared with the physical qubits initialized in known eigenstates of the Pauli-Z (Pauli-X) operator. The world lines of the Majorana modes that move between the distinct boundaries emerge in two well-separated pair creation operations. This is true to the analogy with fault-tolerant quantum computation with Ising anyons.

Figure 15

Lattice surgery between two surface codes. Two surface code lattices are shown in bold colors to the left and the right of the figure. To measure the parity of these two logical qubits, we initialize the qubits that separate the two lattices in known eigenstates of the Pauli-X operator and then begin measuring the stabilizers of the extended rectangular surface code including the pale-colored stabilizers in between the two lattices.

Figure 16

Lattice surgery for the parity measurement ${\overline{Z}}_1{\overline{Z}}_2$ with the foliated surface code. The time axis runs from the bottom of the page to the top. We show two distinct surface codes at the bottom of the page. At the point where we begin the parity measurement the two foliated qubits are connected through an extended piece of foliated surface code. We show the trajectories of the foliated Majorana modes in red. We see that at the connection point two pairs of Majorana modes are fused, where one mode from each pair are taken from the two different foliated codes. Once the parity data is collected the connection is broken by inputing the rectangular lattice into a new channel to separate the two encoded qubits. This is shown at the top of the figure.

Figure 17

Two surface codes where Pauli-X type stabilizers lie on blue faces and Pauli-Z stabilizers on yellow faces. A dislocation is shown in green running through the middle of the right surface code which are the product of two Pauli-X terms and two Pauli-Z terms. An example of such a stabilizer is shown. The left qubit, qubit 1, is initially encoded in an arbitrary state, and the right qubit, qubit 2, is prepared in an eigenstate of the logical Pauli-Z operator. The logical operator whose measurement outcome is inferred from the stabilizer measurements during lattice surgery is shown explicitly.

Figure 18

After transversally measuring the qubits below the defect line of the lattice to the right transversally in the computational basis the right qubit is reduced to a smaller square lattice. The system to the left is initialized in a logical eigenstate of the Pauli-Z operator. The support of the logical parity measurement that is made under the surgery is shown.

Figure 19

The sequence of foliated channels that execute a phase gate on a qubit transmitted with a foliated surface code. Time increases up the page. The worldlines of the Majorana modes are shown in red (green) if they run through the exterior boundary (bulk) of the foliated system and the boundary of the dislocation of the foliated surface code is shown in pink. One can track the world lines of the modes from the bottom to the top of the page to see the two right most world lines exchange.

Figure 20

We show the logical Pauli-X, Pauli-Y, and Pauli-Z operators in panels (a), (b), and (c), respectively. From these diagrams we see that by measuring the appropriate qubit within the time interval in the Pauli-Z basis, and otherwise measuring the preceeding qubits of the system in the Pauli-Y basis, we can measure the logical information of the chain in an arbitrary Pauli basis.

Figure 21

The graph for a type-II foliated qubit where we make a nondestructive logical Pauli-Y measurement at time interval $t$. The blue ancilla qubit is coupled to the chain element indexed $T=Y\left(t\right)$ via a controlled-phase gate. We show the logical Pauli-Y operator that commutes with the single-qubit measurement pattern where the ancilla is measured in the Pauli-X basis and, otherwise, the qubits in the chain are measured in the Pauli-Y basis, as prescribed.

Figure 22

The gauge color code lattice arranged with the qubits lying on the vertices of a cubic lattice with four-colorable cells. (a) The stabilizers are 8 and 32 body terms supported on the cuboidal cells of the lattice. (b) The faces of a yellow cell whose faces are colored with a pair of colors opposite to the colors of the two cells that share the face. (c) The cells are separated to reveal the structure of the lattice more clearly. The gauge terms lie on faces where pairs of cells share common support. (d) A two-dimensional representation of how cells lying on the layer above, marked with bold outlines, lie atop the layer below, where the cells are filled with pale colors.

Figure 23

The stabilizers of a single cell of the resource state for the gauge color code on red qubits indexed $X\left(t\right)$. Measuring the face terms of the gauge color code infers the value of the stabilizer three times. Taking the product of pairs of these cell measurements gives additional stabilizer data for the foliated system.