Equivalence between Fermion-to-Qubit Mappings in two Spatial Dimensions

We argue that all locality-preserving mappings between fermionic observables and Pauli matrices on a two-dimensional lattice can be generated from the exact bosonization in Chen et al. [Ann. Phys. (N. Y) 393, 234 (2018)], whose gauge constraints project onto the subspace of the toric code with emergent fermions. Starting from the exact bosonization and applying Clifford finite-depth generalized local unitary transformation, we can achieve all possible fermion-to-qubit mappings (up to the re-pairing of Majorana fermions). In particular, we discover a new supercompact encoding using 1.25 qubits per fermion on the square lattice. We prove the existence of finite-depth quantum circuits to obtain fermion-to-qubit mappings with qubit-fermion ratios $r=1+1/2k$ for positive integers $k$, utilizing the trivialness of quantum cellular automata in two spatial dimensions. Also, we provide direct constructions of fermion-to-qubit mappings with ratios arbitrarily close to 1. When the ratio reaches 1, the fermion-to-qubit mapping reduces to the one-dimensional Jordan-Wigner transformation along a certain path in the two-dimensional lattice. Finally, we explicitly demonstrate that the Bravyi-Kitaev superfast simulation, the Verstraete-Cirac auxiliary method, Kitaev’s exactly solved model, the Majorana loop stabilizer codes, and the compact fermion-to-qubit mapping can all be obtained from the exact bosonization.

Popular Summary

The duality between fermion and spin systems, so-called fermion-to-qubit mapping, arises from exactly solvable models and plays an essential role in various fields of physics. The exact bosonization is one of the previous methods that utilized the fermionic excitations in the $Z_2$ topological order to realize the duality. This method has a clear physical picture. An interesting question is: are all fermion-to-qubit mappings in two spatial dimensions originating from the exact bosonization? In this paper, we provide an affirmative answer to this question.

Various fermion-to-qubit mappings have distinct Hilbert spaces, so we define the equivalence relation between fermion-to-qubit mappings via entanglement renormalization techniques. In other words, we can apply arbitrary local unitary operators and add or remove any ancillary degree of freedom disentangled from the rest of the system. Based on the theorem that any translationally invariant Pauli stabilizer model of qubits is a finite copy of Kitaev’s surface code, we prove that all fermion-to-qubit mappings are equivalent to the exact bosonization. We further provide explicit quantum circuits that transform other well-known fermion-to-qubit mappings to the exact bosonization.

For practical purposes, the entanglement renormalization techniques enable us to construct all possible fermion-to-qubit mappings from the exact bosonization. For example, we show a new construction with an arbitrarily small qubit-fermion ratio. For the future design of efficient fermion-to-qubit mappings, we can start from the exact bosonization and apply quantum circuits wisely.

Figure 1

We disentangle the green qubit from others by a local unitary transformation $V$ and then remove this part. This is called a generalized local unitary circuit.

Figure 2

Bosonization on a square lattice [2]. We put Pauli matrices $X_e$, $Y_e$, $Z_e$ on each edge and one complex fermion $c_f,c_f^{†}$ at each face. In this figure, faces are labeled a–d and vertices are labeled 1–9. Each edge contains a qubit. We work on the Majorana basis ${\gamma }_f\equiv c_f+c_f^{†}$ and ${\gamma }_f^{\prime }\equiv -i(c_f-c_f^{†})$ for convenience.

Figure 3

We color the faces to “even” and “odd.” Each edge contains a qubit.

Figure 4

The finite-depth Clifford circuit for the $r=1.5$ construction. Here cy denotes the controlled-$Y$ gate. The depicted unitaries are denoted in the sequence as $V_1,V_2,V_3,V_4,V_5,V_6$. The total GLU disentangler is $V_C\equiv V_6{V}_5{V}_4{V}_3{V}_2{V}_1$. After conjugation by unitary operator $V_C$, a part of the stabilizers in Eq. (17) becomes a single Pauli matrix, which can be removed from the system.

Figure 5

The 2d square lattice after disentanglement. The qubits on dashed edges are removed from the system.

Figure 6

The $r=1.5$ construction is the same as the compact fermion-to-qubit mapping [9] after the re-pairing of Majorana fermions above. Each circle represents a complex fermion formed by the two Majorana fermions. The underlying arrows specify the order to form a fermion. The arrows form a Kasteleyn orientation, ensuring that the fermion parity after re-pairing is well defined [40, 41, 42].

Figure 7

The 2d square lattice is colored with four colors: yellow, blue, red, and green. Yellow and red belong to the original “odd” faces, and blue and green belong to the original “even” faces. White dots represent fermionic modes inside faces. Compared with Fig. 5, qubits on the edges between yellow and blue faces will be stabilized by a single Pauli matrix after a certain unitary transformation so that they can be removed from the system. In the end, each solid line has one qubit, while there is no qubit on dashed lines.

Figure 8

The finite-depth Clifford circuit to construct bosonization with $r=1.25$. The depicted unitaries are $V_7,V_8,V_9,V_{10},V_{11},V_{12}$ (from top left to bottom right). The GLU disentangler is $V_{\mathrm{SC}}=V_{12}{V}_{11}{V}_{10}{V}_9{V}_8{V}_7$. After conjugation by the unitary operator $V_{\mathrm{SC}}$, a part of the stabilizers in Eq. (19) becomes a single Pauli matrix, which can be removed from the system.

Figure 9

Shorthand representation of the mapping between fermionic operators and the Pauli matrices. Panel (a) represents the nearest-neighbor horizontal hopping terms. The first diagram indicates that $i{\gamma }_{\mathrm{yellow}}{\gamma }_{\mathrm{blue}}$ is mapped to the Pauli operator $XZZZ$ indicated above. The second diagram (top right) represents that $i{\gamma }_{\mathrm{blue}}{\gamma }_{\mathrm{yellow}}$ is mapped to $-XXY$. The remaining diagrams follow the same rule, e.g., $i{\gamma }_{\mathrm{green}}{\gamma }_{\mathrm{red}}\leftrightarrow XXXZ$ and $i{\gamma }_{\mathrm{red}}{\gamma }_{\mathrm{green}}\leftrightarrow -ZZY$. Panel (b) represents the near-neighbor vertical hopping terms. Note that the fermionic hopping term is $\gamma$ to ${\gamma }^{\prime }$ in this case. For example, the first diagram is $i{\gamma }_{\mathrm{yellow}}{\gamma }_{\mathrm{green}}^{\prime }\leftrightarrow XZ$. Panel (c) represents the fermion parity operators. Each face $f$ indicates the location of the on-site fermionic parity operator $-i{\gamma }_f{\gamma }_f^{\prime }$, which is mapped to the Pauli $ZZZ$ shown in the diagram.

Figure 10

The (potential) flippers. For $G_a^{\prime }$ on the gray face $a$, its flipper is the product of $X$ connecting two gray faces on its right column, shown by the green operator. For $G_b$ on the white face, its potential flipper is the product of $X$ connecting to a gray face below, shown by the blue operator. This potential flipper may anticommute with $G^{\prime }$ on a gray face, which can be fixed by attaching the flipper for this $G^{\prime }$. For $U_{e_1}$ on a horizontal edge $e_1$, the potential flipper is $Z_{e_1}$, which flips exactly one $U_e$ and anticommutes with some $G_f$ and $G_f^{\prime }$ on white and gray faces. This can be fixed by attaching the flippers for these $G_f$ and $G_f^{\prime }$ to the potential flipper of $U_{e_1}$.

Figure 11

The ordering of edges on each vertex. The red numbers are the labels. Note that one edge is connected to two vertices, and the two numbers on the two vertices do not need to be the same. The assigned numbers determine the relative ordering for two edges connecting to the same vertex.

Figure 12

Graph structure of the auxiliary Hamiltonian $H_{\mathrm{aux}}$.

Figure 13

The finite-depth Clifford circuit to convert the Verstraete-Cirac mapping to the exact bosonization. Details of the $H$, $R$, $S$ gates are discussed in Appendix pp2. The finite-depth Clifford circuit that converts the Verstraete-Cirac mapping into the exact bosonization is $V^{\mathrm{VC}}=V_6^{\mathrm{VC}}{V}_5^{\mathrm{VC}}{V}_4^{\mathrm{VC}}{V}_3^{\mathrm{VC}}{V}_2^{\mathrm{VC}}{V}_1^{\mathrm{VC}}$.

Figure 14

To match our exact bosonization to the Verstraete-Cirac mapping, we shift our Majorana modes on each face in the following way: (1) shift ${\gamma }_f^{\prime }$ upward and let it be $\gamma$ on the new face; (2) shift ${\gamma }_f$ rightward and let it be ${\gamma }^{\prime }$ on the new face.

Figure 15

Correspondence of logical operators between the exact bosonization and the Verstraete-Cirac mapping.

Figure 16

Kitaev’s honeycomb model. The red, blue, and green edges represent $x$, $y$, and $z$ links. For each link, the product of two Pauli matrices on its vertices is mapped to the product of $\gamma$ and ${\gamma }^{\prime }$ on its vertices, shown in Eq. (50).

Figure 17

To match our exact bosonization to Kitaev’s honeycomb model, we shift our Majorana modes on each face in the following way: (1) shift ${\gamma }_f$ downward and let it be $\gamma$ on the new face; (2) shift ${\gamma }_f^{\prime }$ leftward and let it be ${\gamma }^{\prime }$ on the new face. We re-pair $\gamma$ and ${\gamma }^{\prime }$ enveloped in the ellipse to form a complex fermion.

Figure 18

The embedding of the honeycomb lattice into the square lattice. The red, blue, and green links correspond to the $x$, $y$, and $z$ types in Kitaev’s honeycomb model.

Figure 19

The finite-depth Clifford circuit for the MLSC to the exact bosonization. The first Clifford circuit will disentangle the qubits on the edges between red and yellow faces, so the edges between red and yellow faces become dashed lines in the second and third steps. The first panel involves three unitary circuits $U_1^{\mathrm{MLSC}},U_2^{\mathrm{MLSC}},U_3^{\mathrm{MLSC}}$ corresponding to the controlled-not (cnot) gates labeled ①, ②, and ③, respectively. The finite-depth Clifford circuit that converts Majorana-loop stabilizer codes to the exact bosonization is $U^{\mathrm{MLSC}}=U_5^{\mathrm{MLSC}}{U}_4^{\mathrm{MLSC}}{U}_3^{\mathrm{MLSC}}{U}_2^{\mathrm{MLSC}}{U}_1^{\mathrm{MLSC}}$.

Figure 20

Horizontal hoppings $i{\gamma }_{L(e)}{\gamma }_{R(e)}$ after a finite-depth GLU transformation in Fig. 19. Labels (a) and (b) denote hoppings between blue and orange dots; labels (c) and (d) denote hoppings between pink and yellow dots. Hoppings (a), (b), and (d) are exactly the horizontal hoppings in the exact bosonization, and hopping (c) is a product of the hopping operator and the stabilizer in the exact bosonization.

Figure 21

Labels (e)–(h) denote vertical hoppings $i{\gamma }_{L(e)}{\gamma }_{R(e)}$ after a finite-depth GLU transformation in Fig. 19. They match the vertical hoppings in the exact bosonization.

Figure 22

The finite-depth Clifford circuit to convert the exact bosonization to the 1d Jordan-Wigner transformation. In the first step, we order the system from left to right, then apply the cnot gate to each column following the above ordering. The cnot gates are applied simultaneously in the second and third steps. In the fourth step, we order the system from right to left, then apply the cnot gate to each column following the right-to-left ordering. In the fifth step, cz gates are simultaneously applied. The depicted unitaries are $U_1^{\mathrm{JW}},U_2^{\mathrm{JW}},U_3^{\mathrm{JW}},U_4^{\mathrm{JW}},U_5^{\mathrm{JW}}$ (from top to bottom). The GLU disentangler is $U_{\mathrm{JW}}=U_5^{\mathrm{JW}}{U}_4^{\mathrm{JW}}{U}_3^{\mathrm{JW}}{U}_2^{\mathrm{JW}}{U}_1^{\mathrm{JW}}$. Note that $U_1^{\mathrm{JW}}$ and $U_4^{\mathrm{JW}}$ are linear-depth local unitary circuits.

Figure 23

Ordering of the 1d Jordan-Wigner transformation on the square lattice.

Figure 24

Physical Hilbert space of the fermion-to-qubit mapping with $r=1+1/l$. Each black site denotes one qubit, and each red site denotes two qubits. Columns of red vertices are separated by a distance $l$.

Figure 25

The physical and logical Hilbert spaces for supercompact encoding. The left-hand side is the physical Hilbert space, where each black vertex contains one qubit, and each gray vertex contains two qubits. The right-hand side is the logical Hilbert space, where each vertex encodes a fermionic mode. The qubit-fermion ratio $r$ is $1.25$ in this setting.

Figure 26

The hopping term ${\tilde{A}}_e$ and the parity term ${\tilde{B}}_v$ in the bosonic Hilbert space. The definitions of ${\tilde{A}}_e$ and ${\tilde{B}}_v$ depend on the colors of edges and vertices. (a)–(d) Four kinds of horizontal hopping terms; (e)–(h) four kinds of vertical hopping terms; (i),(j) parity terms on black and gray vertices.

Figure 27

The stabilizer acts on the vertex $d$ that connects to pink, black, purple, and green edges. This stabilizer comes from identity (C5) for a closed loop $a\to b\to c\to d\to a$. The product of ${\tilde{A}}_e$ on any closed loop is generated by this stabilizer.

Figure 28

Stabilizer constraint for the closed loop $d\to c\to g\to h\to d$, which gives $-I$, and it matches identity (C5) on the fermion side $A_{e_{hd}}{A}_{e_{hg}}{A}_{e_{gc}}{A}_{e_{dc}}=-I$.