Methods for Simulating String-Net States and Anyons on a Digital Quantum Computer

The finding of physical realizations of topologically ordered states in experimental settings, from condensed matter to artificial quantum systems, has been the main challenge en route to utilizing their unconventional properties. We show how to realize a large class of topologically ordered states and simulate their quasiparticle excitations on a digital quantum computer. To achieve this, we design a set of linear-depth quantum circuits to generate ground states of general string-net models together with unitary open-string operators to simulate the creation and braiding of Abelian and non-Abelian anyons. We show that the Abelian (non-Abelian) unitary string operators can be implemented with a constant- (linear-) depth quantum circuit. Our scheme allows us to directly probe characteristic topological properties, including topological entanglement entropy, braiding statistics, and fusion channels of anyons. Moreover, this set of efficiently prepared topologically ordered states has potential applications in the development of fault-tolerant quantum computers.

Popular Summary

Topological phases of matter have been at the forefront of condensed-matter physics since the discovery of the fractional quantum Hall effect. Topological order presents a new paradigm in understanding phases of matter; unlike conventionally ordered phases such as crystals or magnets, it cannot be probed locally and is associated with exotic phenomena such as fractionalization and unconventional statistics of its excitations. A lack of local signatures makes experimental detection very challenging. An alternative approach is to simulate such states using a quantum computer.

We propose unitary quantum circuits implementable on a quantum computer to synthesize string nets, which are archetypal topologically ordered states the excitations of which obey exotic non-Abelian braiding statistics. The depth of these quantum circuits scales linearly with the size of the system, which is optimal for topologically ordered states. Since topological order lacks conventional order parameters, we also propose methods of detection via nonlocal measurements. These include creating and braiding the quasiparticles, setting out the path to experimental detection of their most interesting and sought-after property: non-Abelian anyonic statistics.

We specifically focus on experimentally producing and manipulating Fibonacci anyons, the simplest example of anyons, the braiding of which is computationally universal—using braiding operations alone, one can approximate any sequence of unitary quantum gates. While it might seem silly to use a quantum computer to simulate another quantum computer, it is anything but: the experimental detection of non-Abelian statistics remains a holy grail for condensed-matter physics and a proof-of-principle demonstration of the viability of braiding-based quantum computation will constitute a crucial advance.

Figure 1

The construction of the toric code ground state. (a) The honeycomb lattice with qubits on the edges that contains a plaquette (black) and a vertex (purple): the vertices on the boundary only have two incoming legs. (b) The quantum gate C-${\hat{B}}_p$ that is controlled by the representative qubit and targets the rest. The green marks the control qubits (unchanged upon C-${\hat{B}}_p$), while the yellow marks the target qubits (changed). (c) A graphical illustration of the algorithm: prepare the red qubits in $|0⟩$ and the blue qubits in $|+⟩=(|0⟩+|1⟩)/\sqrt{2}$; then apply C-${\hat{B}}_p$ on each plaquette in parallel for each row.

Figure 2

The construction of the double-semion ground state. We implement the algorithm as an analogy to Fig. 1: the representative qubits (blue) are initialized in $|-⟩=(|0⟩-|1⟩)/\sqrt{2}$ and the rest of the qubits (red) are $|0⟩$. (a) The construction consists of parallel application of C-${\hat{B}}_p$ on each row. (b) The circuit diagram for C-${\hat{B}}_p$. The two-qubit gate symbolized by two solid dots connected by a line is a controlled-$Z$ gate. The three-qubit gate applies ${\hat{\sigma }}^z$ to the third qubit if the solid and hollow control qubits are 1 and 0, respectively; otherwise, it does nothing. The dashed box in the circuit diagram contains the unitary that gives a phase $+1$ ($-1$) if the number of loops changes (unchanged) when a TC plaquette operator is applied [as shown in (c)].

Figure 3

The construction for general string-net models. (a) During the construction, C-${\hat{B}}_p^s$ only acts on the subspace shown on the left; here, $i_1$, $i_2$, $i_3$, $e_1$, $e_2$, $e_3$, and $e_4$ label all the allowed strings under branching rules. $s$ is the state of the representative qudit. The arrows indicate the convention for the string orientation. One can prepare general string-net states on a honeycomb lattice by applying C-${\hat{B}}_p$ row by row. Despite the overlap of the gates, they can be decomposed for parallel implementation on each row. The C-${\hat{B}}_p$ gate is controlled by the green qudits and alters the yellow ones. The representative qudits (blue) are initialized in $\mathcal{D}\sum _i{a}_i|i⟩$. (b) The gate structure of C-${\hat{B}}_p$. If the representative qudit is in state $|s⟩$, it applies ${\hat{B}}_p^{\prime s}$ to the rest of the spins within the plaquette. This gate satisfies Eq. (6). We provide an explicit circuit for the gate in Appendix pp5.

Figure 4

Probing the TEE on a quantum computer. (a) The simply connected domains $A$, $B$, and $C$ are used in the subtraction procedure given in Eq. (7); the exterior is the rest of the system. (b) A connected domain on a honeycomb lattice for extracting the TEE. The domain includes the qudits on the external legs, while the qudits on the boundary of two domains are shared by both sides.

Figure 5

Probing braiding statistics by interferometry. The unitary paths $\hat{V}$ and $\hat{U}$ for obtaining (a) $S_{ab}$ and (b) the twist factor ${\theta }_a$. (c) A simple circuit for measuring the expectation $⟨\psi |\hat{A}|\psi ⟩$, where $\hat{A}$ is a unitary operator acting on $|\psi ⟩$ and the ancilla qubit is initially prepared in $|0⟩$. At the end of the circuit, the ancillary qubit is measured to obtain $⟨{\hat{\sigma }}^z-i{\hat{\sigma }}^y⟩$. The expectation of $\hat{A}$ follows from $⟨\psi |\hat{A}|\psi ⟩=⟨{\hat{\sigma }}^z-i{\hat{\sigma }}^y⟩$. (d) An alternative path to measure the $S$ matrix. The path can be used to measure an unknown anyon (blue) at a fixed position by braiding with anyons of known species (black).

Figure 6

Preparing and moving anyons with unitary string operators. (a) Open-string operators for Abelian (left) and non-Abelian (right) anyons. The operators act on the set of labeled qudits $\{i\}$ along the path and $\{e\}$ of the legs. Only the states of the qudits $\{i\}$ along the path are changed by the operator. Each quasiparticle has an associated vertex (red) and plaquette (yellow). Before the string operators are applied, the ancillary qudits in the non-Abelian case are initialized to align with the other qudits on the same edge. (b) The endpoint quasiparticles can be moved by extending the string operator sequentially with unitary gates. The dark gray marks the existing string. The enclosed qudits on the left are acted on by the endpoint-moving unitary gates, which correspond to an exact (modified) form of the unitary string operators for the Abelian (non-Abelian) case. The plaquette-vertex labels are used to locate the quasiparticle. When extending the string operator, they trace the motion of the quasiparticle. Note that the ancillary qudit in the non-Abelian case can be moved without physically displacing the qudits. For more details, see Appendices pp2 and pp5.

Figure 7

The string operators for toric code. The end points can be labeled by a plaquette or vertex that corresponds to exactly the microscopic excitation in Eq. (1). The dashed line indicates the path that the plaquette excitation takes as the Wilson string is generated.

Figure 8

The manipulation of Abelian and non-Abelian anyons. (a) Two separate Abelian anyons $a$ and $\overline{a}$ can be created by a sequence of multiple local anyon-antianyon pairs. The intermediate quasiparticles fuse to the vacuum. (b) If $a$ and $\overline{a}$ are non-Abelian anyons, the transport of the anyon is no longer equivalent to local pair creation and pair annihilation of $a$ and $\overline{a}$. The fusion yields a superposition of the vacuum and some other nontrivial anyon species $b$, with amplitudes $p_b$.

Figure 9

The decomposed string operators using the movements from Fig. 6. (a) For the Abelian string operators, an initial short string (dark gray) can be extended to a long string operator (light gray) by a constant-depth local quantum circuit. (b) A sequentially applied quantum circuit is required for extending a short non-Abelian string, resulting in a linear-depth quantum circuit. We show the details of the circuits, including the short string initialization, in Appendix pp2.

Figure 10

Small-scale experimental realization. (a) In a three-plaquette toric code, the TEE can be measured using the three connected domains as in Fig. 4, consisting of three qubits. On the right, we show the initial configuration of the anyons prepared by ${\hat{\sigma }}^x$ and ${\hat{\sigma }}^z$: the anyons can also be moved using the same operators. (b) Braiding steps to realize the exchange of $e\times m$ particles. The solid lines are string operators used to drag the particles. (d) We can extract other $S$-matrix elements by following (c) twice with some specific anyons removed. Here, we only keep the $m$ on the left and the $e$ on the right. Applying (c) twice results in three parallelized steps as shown, which gives $S_{em}=S_{me}=-1$. (d) The braiding for general Abelian and non-Abelian anyons can be simulated on a small system as shown. The left and right diagrams depict the interfered paths of the anyons. Note that the final anyon configurations are identical.

Figure 11

(a) Inverting a string $s$ is the same as changing to the dual string $s^{\ast }$. (b) The orientation convention for branching at a vertex.

Figure 12

A local change of registered connectivity, to make sure that the endpoint of the string can land at an edge with two qudits. No physical gates are needed here.

Figure 13

The quantum circuits for Abelian open-string operators. (a) A circuit realization for preparing Abelian anyons. The shaded box shows the $F$ move defined by the $F$ symbol in the Abelian model, together with the labeling convention for the $F$ symbol. We define the shaded circuit element to associate the qudit state with an $F$-symbol phase. This phase is computed based on the $F$-symbol labels assigned in the circuit diagram (see also the discussion in Appendix pp5). (b) A circuit realization for moving the Abelian anyons by extending the string operator (dark gray).

Figure 14

The quantum circuits for the non-Abelian open-string operators. The orange color highlights the qudits that store the ancillary qudit states. (a) A circuit realization for preparing non-Abelian anyons. The shaded box shows the $F$ move defined by the $F$ symbol in the non-Abelian model, together with the labeling convention for the $F$ symbol. The shaded circuit elements are qudit quantum gates that perform the $F$ move. The gates are controlled single-qudit rotations that change the state of the green qudit labeled with $m$. The gates are controlled by the states of the other input qudits, each of which is assigned with a label in the $F$ symbol according to the circuit diagram. (b) A circuit realization for moving the non-Abelian anyons by extending the open-string operator (dark gray).

Figure 15

The definition for the qudit gates used in the construction of the open-string operators, where $s$ is the label for the corresponding simple string operator. The quantity ${\omega }_{p^{\prime },p}$ is the phase ${\omega }_k$ in Eq. (A9), with $i_k^{\prime }=p^{\prime }$ and $i_k=p$, and the orientation is to be identified from the direction of the string operators in the circuit diagrams, such as in Figs. 13 and 14.

Figure 16

A circuit decomposition for the C-${\hat{B}}_p$ gate using the labeling convention in Eq. (E1). All the gate operations are controlled by the qudit on the top bond (the green $s$). The dashed arrows show the virtual location of the ancillary qudit state, which is stored in the initially disentangled qudit (orange) throughout the steps. In step 1, the qudit gate is a qudit cnot followed by Fig. 14. The cnot aligns the $|0⟩$ qudit (orange) with the qudit at the endpoint of the string operator that is to be initialized. In steps 2 and 3, the qudit gate is the same as Fig. 14, followed by a swap gate that keeps the ancillary qudit state in the orange qudit. Step 4 is similar to steps 2 and 3, except that we adapt the circuit slightly, since no ancillary qudit is required at the last step. The explicit circuit diagrams are given in Fig. 17.

Figure 17

Explicit circuit diagrams for Fig. 16. The qudits are labeled by Eq. (E1). (a) The C-${\hat{B}}_p$ for the Abelian case. Here, we use the controlled fusion gate C-$S$, which performs the unitary C-$S|s,i⟩=|s,s\times i⟩$. (b) The circuit for the non-Abelian case; the qudit used to store the ancillary state is marked in orange.

Figure 18

A quantum circuit for the $F$ symbol given in Ref. [65], where all the gates are qubit gates. Here, we use a four-qubit controlled-rotation gate, i.e., a single-qubit rotation is applied to the fifth qubit if the qubits at the solid dots are $|1⟩$; otherwise, no operation will be performed. The single-qubit rotation is defined in Eq. (F1).