Quantum simulation of lattice gauge theories via deterministic duality transformations assisted by measurements

Quantum simulation is one of the major applications of quantum devices. In the noisy intermediate-scale quantum era, however, the general quantum simulation is not yet feasible, such as that of lattice gauge theories, which is likely limited due to the violation of the Gauss law constraint and the complexity of the real-time dynamics, especially in the deconfined phase. Inspired by the recent works of Ashkenazi and Zohar [Phys. Rev. A 105, 022431 (2022)] and of Tantivasadakarn, Thorngren, Vishwanath, and Verresen [arXiv:2112.01519], we propose to simulate dynamics of lattice gauge theories by using the Kramers-Wannier transformation via cluster-state-like entanglers, mid-circuit measurements, and feedforwarded corrections, which altogether is a constant-depth deterministic operation. In our scheme, specifically, we first quantum simulate the time evolution under a corresponding symmetric Hamiltonian from an initial symmetric state, and then apply the Kramers-Wannier procedure. This results in a wave function that has time evolved under the corresponding lattice gauge theory from a corresponding initial, gauged wave function. In the presence of noises in time evolution, the procedure succeeds when we are able to pair up magnetic monopoles represented by nontrivial measurement outcomes. Furthermore, given a noise-free Kramers-Wannier transformation, the resulting wave function from a noisy time evolution satisfies the Gauss law constraint. We give explicit examples with the low dimensional pure gauge theories and gauge theories coupled to bosonic or fermionic matters such as the Fradkin-Shenker model.

Figure 1

The procedure of the KW transformation of time evolution. (Top) A quantum circuit expression of the result. Starting from an ungauged wave function, we perform a time evolution $T\left(t\right)$ via a quantum circuit. The Kramers-Wannier transformation procedure consists of entangling the original degrees of freedom with additional degrees of freedom by entanglers and measuring the original degrees of freedom. We obtain a gauged wave function evolved with a time evolution $T^{*}\left(t\right)$ with a Hamiltonian obtained by the Kramers-Wannier duality, up to a byproduct operator ${\mathcal{O}}_{\text{bp}}$, which depends on the measurement outcomes.

Figure 2

The measurement-assisted Kramers-Wannier transformation in a one-dimensional periodic lattice. (i) A state $|{\psi }_{\text{in}}\rangle$ is initialized on vertices (black dots) of the one-dimensional (1d) chain, and is evolved under the Hamiltonian of the TFI model. Ancillary qubits ${|0\rangle }^{\otimes E}$ are placed on edges (blue dots) separately. (ii) We apply the controlled-$X$ gate, where the vertex qubits control $X$ gates on edge qubits, as indicated by the arrows in the figure. (iii) We measure the vertex qubits in the $X$ basis. When the measurement outcome is $X=-1$ (occurring at check marks and red dots below), phase operators effectively act on the desired dual time evolution. It can be expressed as ${\mathcal{O}}_{\text{bp}}$, the product of $Z$ operators on edge qubits that overlap with purple lines which connect red dots. (iv) One can negate the byproduct operator by applying the counter operator ${\mathcal{O}}_{\text{counter}}$, the product of $Z$ operators on edge qubits marked by orange lines that connect red dots. As ${\mathcal{O}}_{\text{bp}}\times {\mathcal{O}}_{\text{counter}}$ is trivial on the state after the map, we obtain the dual time evolution which is not affected by randomness of measurement outcomes.

Figure 3

An example of 1-chain $c_1$, which defines a basis $|c_1\rangle$ in the Hilbert space. The solid lines correspond to one, while ordinary thin lines correspond to zero.

Figure 4

An example of the intersection pairing number $\#(c_1\cap c_1^{*})$ between a 1-chain $c_1$ and a dual 1-chain $c_1^{*}$.

Figure 5

The two-dimensional (2d) square lattice and its dual. The red boxes represent the operators that appear in the Hamiltonian $H^{\text{TFI}}$ and the blue ones in the Hamiltonian $H^{\text{GT}}$ and the generator of the gauge transformation.

Figure 6

(left) The ungauged basis $|c_0\rangle$ is a tensor product of $Z$ basis vectors whose eigenvalues are coefficients in the expansion of 0-chain $c_0$ with the 0-cell basis. A 0-chain is naturally identified with a dual 2-chain. (right) The gauged basis $|{\partial }^{*}{c}_0\rangle$ is a tensor product of bits living on edges, each of which is a sum of bits on vertices surrounding it.

Figure 7

Graphic explanation of Eq. (65). (a) On vertices, we have the time-evolved wave function, $T^{\mathrm{TFI}}\left(t\right)|{\psi }_{\mathrm{ungauged}}^{\left(0\right)}\rangle$. We introduce the ancillary product state $|0\rangle$ to the edges of the lattice. (b) We apply the entangler ${\mathcal{U}}_{\mathrm{KW}}$ which consists of the $CX$ gates whose controlling qubits are on vertices and target qubits are on edges. (c) We measure the vertex degrees of freedom with the $X$ basis. (d) The byproduct operator ${\mathcal{O}}_{\mathrm{bp}}$ is a product of Pauli $Z$ operators supported on a set of paths ${\rho }_1$, whose endpoints are at the vertices with $s_v=1$.

Figure 8

An example of the byproduct operator ${\mathcal{O}}_{\text{bp}}$ resulting of the randomness of measurements. The vertices $v_1$ and $v_2$ has nontrivial measurement outcomes: $s_0=v_1+v_2$. The Pauli $Z$ operator is applied along the path ${\rho }_1$ such that $\partial {\rho }_1=s_0$, depicted with purple lines. Magnetic monopoles are induced at these two vertices (dual plaquettes). We apply a counter phase operator ${\mathcal{O}}_{\text{counter}}$ along the path ${\tau }_1$ (orange) such that it ends at $v_1$ and $v_2$; $\partial {\tau }_1=s_0$. The resulting phase operator ${\mathcal{O}}_{\text{bp}}\times {\mathcal{O}}_{\text{counter}}$ is the product of $G_{{\sigma }_0^{*}}$ over dual vertices inside $\mathcal{R}$.

Figure 9

The byproduct operator $Z\left({\rho }_1^{\left(i\right)}\right)$ ($i=1,2$) and the counter operator $Z\left({\tau }_1\right)$ on a torus.

Figure 10

(a) The ${\mathcal{O}}_v$ term in the Hamiltonian (86). (b) The first term (plaquette term) in the twisted gauge theory Hamiltonian (89). (c) The Gauss law operator in the twisted gauge theory.

Figure 11

(a) The action of the entangler in the computational basis. The symbol $a_u$ is a ${\mathbb{Z}}_N$ integer. (b) The pattern of the entangler. The controlled-$X$ gate acts as ${\left[C{X}_{{\sigma }_0,{\sigma }_1}\right]}^{a({\partial }^{*}{\sigma }_0;{\sigma }_1)}$, where $a({\partial }^{*}{\sigma }_0;{\sigma }_1)=a({\partial }^{*}{\sigma }_2^{*};{\sigma }_1^{*})=\pm 1$ (or 0) is determined according to the orientation of dual edges relative to the boundary of the dual plaquette, as explained in Eq. (101). The basis $|c_0,0_1\rangle$ is mapped to $|c_0,{\partial }^{*}{c}_0\rangle$ as before. (c) The entangler transforms the $X_{{\sigma }_0}$ term in the clock model to the product of the five Pauli $X\text{s}$ in the figure. The blue square is the plaquette term in the gauge theory, i.e., $X\left({\partial }^{*}{\sigma }_2^{*}\right)$. (d) The Gauss's law operator in the gauge theory, $Z\left(\partial {\sigma }_2\right)=Z\left(\partial {\sigma }_0^{*}\right)$. This is a symmetry in the image of the Kramers-Wannier map ${\widehat{\text{KW}}}^{{\mathbb{Z}}_N}$: $Z\left(\partial {\sigma }_2\right)|{\partial }^{*}{c}_0\rangle ={\omega }^{\#(\partial {\sigma }_2\cap {\partial }^{*}{c}_0)}|{\partial }^{*}{c}_0\rangle =|{\partial }^{*}{c}_0\rangle$.

Figure 12

The procedure of obtaining the time evolution with the gauged Ising (left), Majorana fermion (right) model. (a) The ungauged wave function placed on vertices (black dots) is evolved under the Hamiltonian of the transverse and longitudinal Ising model. Separate degrees of freedom are prepared on the dual chain as a product state. On the left, to obtain the gauged Ising model, the product state is placed on both dual vertices ($v^{*}$: blue circle) and dual edges ($e^{*}$: blue square). On the right, to obtain the gauged Majorana fermion model (${\mathbb{Z}}_2$ QED), the fermionic vacuum state is placed on dual vertices ($v^{*}$: orange circles) while the bosonic product state is placed on dual edges ($e^{*}$: blue squares). (b) We apply the entanglers. For the gauged Ising model, we apply $CX$ gates, according to the arrows that point from controlling qubits to target qubits. For the ${\mathbb{Z}}_2$ QED, we apply $CX$ and $CS$ gates, where the edges are oriented to the right as a convention. (c) We measure the original degrees of freedom (black dots), resulting in a time-evolved gauged wave function on the dual chain (blue and orange dots) up to byproduct operators. Circles are the matter field while squares represent the gauge field.

Figure 13

The red boxes represent the operators that appear in the Hamiltonian $H^{\text{SP}}$ and the blue ones in the Hamiltonian $H^{\text{FS}}$ and the generator of the gauge transformation.

Figure 14

The entangler used to implement the Fradkin-Shenker map.

Figure 15

The left two columns depict the conjugation of the $X$ operators in Eqs. (156) and (157) via the entangler ${\mathcal{U}}_{\text{FS}}$ (top two rows), and the operator resulting after the measurement (bottom two rows). The right two columns show the operator map for the $Z$ operators; see Eqs. (159) and (160).

Figure 16

An interpretation of the relation between the $(2+1)\mathrm{d}$ TFI and the gauge theory in terms of the corresponding three-dimensional (3d) Euclidean lattice theories [1]. A discussion on the mobility of the Kramers-Wannier duality defect or operator is given in Refs. [93, 94, 95], for example