Quantum circuits for toric code and X-cube fracton model

2025-05-02 0 0 717.58KB 23 页 10玖币

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Quantum circuits for toric code and X-cube fracton

model

Penghua Chen1, Bowen Yan1, and Shawn X. Cui1,2

1Department of Physics and Astronomy, Purdue University, West Lafayette

2Department of Mathematics, Purdue University, West Lafayette

We propose a systematic and eﬃcient quantum circuit composed solely of

Cliﬀord gates for simulating the ground state of the surface code model. This

approach yields the ground state of the toric code in ⌈2L+2+log2(d)+ L

2d⌉time

steps, where Lrefers to the system size and drepresents the maximum distance

to constrain the application of the CNOT gates. Our algorithm reformulates

the problem into a purely geometric one, facilitating its extension to attain the

ground state of certain 3D topological phases, such as the 3D toric model in

3L+ 8 steps and the X-cube fracton model in 12L+ 11 steps. Furthermore, we

introduce a gluing method involving measurements, enabling our technique to

attain the ground state of the 2D toric code on an arbitrary planar lattice and

paving the way to more intricate 3D topological phases.

1 Introduction

The subject of topological phases of matter (TPMs) has been under extensive study for the

past few decades. Topological phases are gapped spin liquids at low temperatures which

are not described by the conventional Landau theory of spontaneous symmetry breaking

and local order parameters; instead, they are characterized by a new order, topological

order. The ground states of a topological phase have stable degeneracy and robust long

range entanglement. Topological phases in 2D also support quasi-particle excitations with

anyonic exchange statistics which make them an appealing platform to fault-tolerantly

store and process quantum information. Two peculiar features among others are that

the ground state degeneracy is a topological invariant of the underlying system, and that

the quasi-particles can freely move without costing energy. A large class of topological

phases is realized by exactly solvable spin lattice models with bosonic degrees of freedom.

A paradigmatic example in 2D is the toric code, and more generally Kitaev’s quantum

double model based on ﬁnite groups [6,10], and yet even more generally the Levin-Wen

string-net model based on fusion categories [11]. Examples of 3D topological phases include

3D toric model and the Walker-Wang model based on premodular categories [23].

In recent years, more exotic phases in 3D, called fracton phases, have been discovered

[8,21,22]. Fractons also possess stable ground state degeneracy and long range entangle-

ment. However, the ground state degeneracy of fractons depends on the system size, and

hence is not a topological invariant. Moreover, the mobility of excitations is constrained.

Penghua Chen: chen3014@purdue.edu

Bowen Yan: yan312@purdue.edu, The ﬁrst two authors contributed equally to this work.

Shawn X. Cui: cui177@purdue.edu, Corresponding author.

Accepted in Quantum 2024-02-28, click title to verify. Published under CC-BY 4.0. 1

arXiv:2210.01682v3 [cond-mat.str-el] 29 Feb 2024

The excitations can only move in certain subsystems or cannot move at all. Well known

examples of fractons include the Haah code [8] and the X-cube model [22]. While regular

topological phases are described by topological quantum ﬁeld theories, it is still an open

question what theories mathematically characterize fractons. Since fractons also satisfy

the topological order conditions in the sense of [2], we call the ground states of a fracton

topologically ordered states, in the same way as those of regular topological phases.

Realizing topological phases in physical systems remains an extremely challenging task.

On the other hand, there now exist quantum processors based on a number of platforms

such as superconducting qubits [14], Rydberg atomic arrays [7], etc. These devices can

host physical qubits at the scale of 102, and this number is expected to increase signiﬁ-

cantly in the near future. Hence, it is both feasible and interesting to simulate topological

phases in quantum processors. Thanks to the intrinsic robustness of topological phases,

the simulation is relatively less sensitive to the noises in the current quantum processors.

We may also gain more insight in topological phases by engineering them in processors.

The toric code ground states were realized in the superconducting-qubit-based systems

[14] and the Rydberg-atom systems [19]. In [14], the authors gave a quantum circuit

consisting of Cliﬀord gates to realize the ground states of the planar toric code (a.k.a.

surface code [5]). Quantum circuits realizing non-Abelian topological orders such as Levin-

Wen string-net model and Kitaev quantum double model have also been studied. See for

instance [12,18,20,4,16,17], though in these cases, the gates utilized are no longer in

the Cliﬀord group and measurements are required.

In this paper, we develop quantum circuits realizing the ground states for a number of

topological phases. In [14], only planar toric code is considered where the lattice is deﬁned

on a planar surface. Here we generalize their method to apply to a large class of surfaces

with or without boundary. The quantum circuit consists of only Cliﬀord gates. In toric

code, the Hamiltonian consists of two types of operators, the term Avfor each vertex v

and the term Bpfor each plaquette p. See Figure 2. The key idea of constructing the

ground state in [14] is as follows. Start with the product state |ϕ0⟩=|00...0⟩(also written

as |0⟩⊗) which is the +1 eigenstate for all vertex terms. The ground state is then obtained

by projecting |ϕ0⟩to the +1 eigenstate of all plaquette operators,

|GS⟩ ∼ Y

1 + Bp

2|ϕ0⟩.(1)

The eﬀect of 1+Bp

2acting on certain states can be simulated by an appropriate combination

of the Hadamard gate and the CNOT gate. For this method to work, the control qubit

for CNOT has to be in the |0⟩state prior to applying the Hadamard and CNOT. Hence,

it is critical to choose the right sequence for the plaquettes so that, immediately before

simulating the term corresponding to each plaquette p, there is always at least one edge

on the boundary of pwith the state |0⟩. When the lattice is a simple planar lattice, the

problem can be easily solved by dividing the lattice into several parts and applying the

CNOT gates in a speciﬁc order. In this paper, since we consider lattices on arbitrary

surfaces, this question is much subtler.

Here we provide an explicit algorithm to determine the sequence in which the plaquette

operators are simulated. We show that this is always possible for a large class of lattices

with or without boundary. The result of the algorithm is a quantum circuit consisting of

Cliﬀord gates realizing the ground state of the toric code. Moreover, we also adapt this

method to 3D phases including the 3D toric model and the X-cube fracton model. For

the X-cube model, we again initialize the state to the product of |0⟩state and simulate

the projectors corresponding to cube terms. A similar issue arises that we need to choose

Accepted in Quantum 2024-02-28, click title to verify. Published under CC-BY 4.0. 2

the correct sequence to simulate the cube terms. We note that the circuit we provide here

realizes an exact ground state of the X-cube model. By comparison, using cluster states

and measurements, the authors in [20] gave an approximate realization of the model.

In addition to the above method using only quantum gates, we also provide a diﬀerent

way of realizing the same states. The alternative way, called gluing method, combines

Cliﬀord gates and measurement of the Pauli Xgate. The resulting circuit has a shorter

depth than the ﬁrst one. Of course, for the toric code or X-cube model, it is possible to only

use measurement to obtain the ground state. Considering that frequent measurements in

near-term quantum processors are costly, our method is a trade-oﬀ between circuit depth

and degree of measurements.

2 Realizing ground state of 2D toric code

2.1 Toric code

It is well known that for any Hamiltonian of the form

H=−X

Pi,(2)

where all elements in Piare projectors and mutually commuting, |GS⟩as deﬁned below is

a ground state as long as it is non-zero:

|GS⟩=Y

Pi|ϕ⟩,(3)

where |ϕ⟩stands for an arbitrary state. Speciﬁcally, in a given connected lattice Γ,Vrefers

to the set of vertices, Prefers to the set of plaquettes and Erefers to the set of edges. We

deﬁne Bo(p)⊆E, p ∈Pto be the set of boundary edges of the plaquette p,τ(e)⊆P, e ∈E

to be the set of plaquettes for which eis a boundary edge, and σ(e)∈V, e ∈Eto be the set

of vertices attached with the edge e. When each edge is associated with a Hilbert space,

we may abuse the notation and use eto represent both an edge and the Hilbert space

associated to the edge. As an example, if an edge eappears as a subscript of an operator,

it means the operator acts on the Hilbert space attached to the edge e.

z-boundary

x-boundary

direct string

x-boundary

z-boundary

dual string

Figure 1: The black solid net on the left represents the lattice Γand the black dashed net on the right

represents the dual of Γinduced by the gray net.

As shown in Figure 1, an edge eis a z-boundary when τ(e)contains only one element,

and it is an x-boundary if σ(e)contains only one element (see [5] for details). On the

lattice Γ, a direct string Sis a series of edges ei,i= 1 ···nsuch that τ(ei)Tτ(ei+1)̸=⊘

for 1≤i<n. A direct string operator F(S)is one operator applies Xon all edges along

Accepted in Quantum 2024-02-28, click title to verify. Published under CC-BY 4.0. 3

the string S, thereby generating two electric charges at both ends. Likewise, the concept

of a dual string S′can be introduced, representing a direct string in the dual lattice of

Γ. A dual ribbon operator F(S′)applies Zto all edges intersected by the dual string S′,

resulting in the creation of two magnetic charges at its endpoints. It is worth noting that

dual string operators that encounter a z-boundary at one end will generate (or annihilate)

a magnetic charge at the other end.

XX Bp

Figure 2: Deﬁnitions of Avand Bpoperators in toric code.

The toric code Hamiltonian His composed of operators deﬁned in Figure 2:

H=−X

v∈V

Av−X

p∈P

Bp.(4)

The action of Av(vertex term) is to apply Pauli matrix Zover edges eif v∈σ(e), and

Bp(plaquette term) acts to apply Pauli matrix Xover edges eif p∈Bo(e). Given that

v=B2

p= 1 and [Av, Bp] = 0 for all v∈Vand p∈P, it can be easily veriﬁed that 1+Av

and 1+Bp

2function as projectors. By substituting Avwith 1+Av

2and Bpwith 1+Bp

2in the

Hamiltonian, we achieve a equivalent form that matches the form of Equation 2. This is

due to a natural one-to-one correspondence between their spectra. So we get a ground

state 1by Equation 3:

|GS⟩=Y

p∈P

1 + Bp

2|ϕ0⟩,(5)

where |ϕ0⟩=|00...0⟩represents a product state where each qubit is in the |0⟩state, and we

drop all 1+Av

2s due to its trivial action on |ϕ0⟩. This state is non-zero due to the presence

of positive coeﬃcients of each components.

2.2 Single plaquette

To systematically introduce our ground state simulation method, we initiate with the most

elementary scenario: applying 1+Bp

2on a single plaquette, which is the basic structure in

2D toric code. A Hadamard gate His naturally described by X+Z

√2, and CNOT gate Ci→j

is deﬁned as

Ci→j|ij⟩= (1−Zi

2Xj+1 + Zi

2)|ij⟩,(6)

where iis the control qubit and jis the target qubit.

In the single plaquette shown in Figure 3, four qubits labeled 1,2,3,and 4are initialized

to the state |0⟩. Subsequently, we will systematically implement Hadamard and CNOT

1The ground state degeneracy of the 2D toric code on a torus is four, and the state |GS⟩corresponds

to |00⟩. A comprehensive explanation can be found in Section 2.5.

Accepted in Quantum 2024-02-28, click title to verify. Published under CC-BY 4.0. 4

4C1→2

|0⟩+|1⟩

|0⟩

|00⟩+|11⟩

2|0000⟩+|1111⟩

Figure 3: Initially, a qubit in the state |0⟩is situated at each gray dot. As quantum gates are applied

to these qubits, their color changes to black. A circle positioned on a dot signiﬁes the application of a

Hadamard gate to the corresponding qubit, while an arrow indicates a CNOT gate, with the arrowhead

pointing from the control qubit to the target qubit.

gates in a speciﬁc sequence, as outlined in the ﬁgure. After the application of H1and

C1→2, we have

C1→2H1|0000⟩= (1−Z1

2X2+1 + Z1

2)X1+Z1

√2|0000⟩=X1X2+ 1

√2|0000⟩.(7)

Explicitly, we can insert a 1+Z

2into the equation as 1+Z

2|0⟩=|0⟩:

(1−Z1

2X2+1 + Z1

2)X1+Z1

√2

1 + Z1

2(8)

√2{(X1

1 + Z1

2X2+X1

1−Z1

2)+(1−Z1

2X2+1 + Z1

2)}1 + Z1

2(9)

√2(X1X2+ 1)1 + Z1

2(10)

Notice 1+Z1

2survives within { } in Equation 9. After reverting to the original expression

and substituting 1+Z1

2with 1, we veriﬁed the accuracy of Equation 7. Importantly, this

equation holds for any quantum state |ϕ⟩:

C1→2H1|0⟩⊗|ϕ⟩=X1X2+ 1

√2|0⟩⊗|ϕ⟩,(11)

since the key step only requires that the initial state must be the eigenstate of Z1with an

eigenvalue +1. Finally, applying the other CNOT gates results in

i=2

C1→iH1|0000⟩=X1X2X3X4+ 1

√2|0000⟩=1 + Bp

√2|0000⟩,(12)

which is the desired ground state. It is important to observe that this procedure remains

eﬀective as long as a qubit from Bo(p)is initially in the state |0⟩. We term such qubits

as free qubits, and their presence is pivotal when considering scenarios involving multiple

plaquettes.

2.3 Developing to a surface with boundary

Given a complicated lattice Γin the state |ϕ0⟩, we need to ﬁnd a path (termed permissible

order in [12]) through all plaquettes pi, such that Sipi=P, using a sequence of edges

ei∈Bo(pi)where ei/∈Si−1

j=1 pj. Each eiis then utilized as a free qubit to apply the

introduced basic structure, resulting in the accumulation of Qi

1+Bpi

√2over |0···0⟩, which

represents the ground state of the toric code on lattice Γ. To illustrate the procedure, we

take four plaquettes as an example depicted in Figure 4. A path featuring four free qubits

e1to e4has been chosen, where eistarts in the state |0⟩at the onset of every step. Upon

completing the path, the desired ground state is eventually obtained.

Accepted in Quantum 2024-02-28, click title to verify. Published under CC-BY 4.0. 5

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QuantumcircuitsfortoriccodeandX-cubefractonmodelPenghuaChen1,BowenYan1,andShawnX.Cui1,21DepartmentofPhysicsandAstronomy,PurdueUniversity,WestLafayette2DepartmentofMathematics,PurdueUniversity,WestLafayetteWeproposeasystematicandefficientquantumcircuitcomposedsolelyofCliffordgatesforsimulatingthegrou...

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Quantum circuits for toric code and X-cube fracton model

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