Time evolution of uniform sequential circuits

Simulating time evolution of generic quantum many-body systems using classical numerical approaches has an exponentially growing cost either with evolution time or with the system size. In this work we present a polynomially scaling hybrid quantum-classical algorithm for time evolving a one-dimensional uniform system in the thermodynamic limit. This algorithm uses a layered uniform sequential quantum circuit as a variational Ansatz to represent infinite translation-invariant quantum states. We show numerically that this Ansatz requires a number of parameters polynomial in the simulation time for a given accuracy. Furthermore, this favorable scaling of the Ansatz is maintained during our variational evolution algorithm. All steps of the hybrid optimization are designed with near-term digital quantum computers in mind. After benchmarking the evolution algorithm on a classical computer, we demonstrate the measurement of observables of this uniform state using a finite number of qubits on a cloud-based quantum processing unit. With more efficient tensor contraction schemes, this algorithm may also offer improvements as a classical numerical algorithm.

Figure 1

(a) Circuit representing $\langle {\psi }_{\text{R}}\left(\mathit{\theta }\right)\left|\widehat{\mathcal{O}}\right|{\psi }_{\text{L}}\left({\mathit{\theta }}^{\prime }\right)\rangle$ with $|{\psi }_{\text{L}}\left({\mathit{\theta }}^{\prime }\right)\rangle$ and $|{\psi }_{\text{R}}\left(\mathit{\theta }\right)\rangle$ being the same state in left and right representations. The shaded region singles out the repeated circuit element, i.e., the transfer matrix. (b) Decomposition of a state unitary into $M_U$ layers of sequential two-qubit gates. (c) Transfer matrix ${\widehat{T}}_{\left\{ab\right\}}^{\left\{AB\right\}}(\mathit{\theta },{\mathit{\theta }}^{\prime })$ between the left and right representations $|{\psi }_{\text{L}}\left({\mathit{\theta }}^{\prime }\right)\rangle$ and $|{\psi }_{\text{R}}\left(\mathit{\theta }\right)\rangle$. (d) Circuit representation of $\langle l,0\left|{\widehat{U}}_{\text{R}}^{†}\widehat{\mathcal{O}}{\widehat{U}}_{\text{L}}\right|0,r\rangle$ on finite number of qubits. (e) Decomposition of an environment unitary into $M_E$ layers of sequential two-qubit gates.

Figure 2

(a) Quantum circuit for the overlap between the time-evolved left-represented state $\widehat{u}\left(\delta t\right)|{\psi }_{\text{L}}\left({\mathit{\theta }}_t\right)\rangle$ at time $t$ and right-represented state $|{\psi }_{\text{R}}\left({\mathit{\theta }}_{t+\delta t}\right)\rangle$ at time $t+\delta t$. For illustration, here $N_q=3$ is chosen. The orange shaded area singles out the transfer matrix. (b) The explicit form of the transfer matrix ${\widehat{T}}_{\left\{abc\right\}}^{\left\{ABC\right\}}({\mathit{\theta }}_t,{\mathit{\theta }}_{t+\delta t})$. The capital indices $\left\{ABC\right\}$ form a composite out index, similarly $\left\{abc\right\}$ form a composite in index. The arrow direction indicates the operation order (time flow). (c) The circuit of the generalized functional $\mathcal{L}$. The unitary $\widehat{W}\left(\mathit{\beta }\right)$ acts on the first $2N_q-1$ qubits of the $|00...0\rangle$ state prepared on $2N_q$ qubits, then unitary $\widehat{O}(\mathit{\theta },{\mathit{\theta }}^{\prime })$ acts on all qubits, and ${\widehat{V}}^{†}\left(\mathit{\alpha }\right)$ acts on the last $2N_q-1$ qubits. (d) General Hadamard test scheme for measurement of the algebraic value of $\mathcal{L}$.

Figure 3

Simulations of time evolution using l-USC with $M_U=1$ and exact environment for the Hamiltonian in Eq. (8) with $g/J=1.0, h/J=0$ (upper row) and $h/J=1$ with $M_U=1$ and 4 (lower row). (a) Expectation value of $\langle {\widehat{\sigma }}^z\left(t\right)\rangle$ using l-USC with different values of $N_q$. Inset: The difference in the single-site density matrices between the quasi-exact iTEBD simulation and optimization of l-USC. (b) The number of parameters required to reach time $t^{*}$ with the fidelity density at least $\mathcal{F}=1-{10}^{-4}$. The black crosses represent the standard iTEBD approach with the black dashed line showing an exponential fit. The blue markers show the results for l-USC, and the line shows the linear fit. Left inset: The reachable time $t^{*}$ under the condition $\mathcal{F}⩾1-{10}^{-4}$ as a function of $N_q$. The dashed line shows a linear fit. Right inset: The error in fidelity density $1-\mathcal{F}$ for different $N_q$. (c) Entanglement entropy as a function of evolution time $tJ$ compared with the quasi-exact result. The horizontal dashed lines mark the theoretical maximum entanglement entropy levels $(N_q-1)log2$. (d)–(f) Same results for the $g/J=h/J=1$ case. In the right inset of (e), linear fits start at $N_q=2$ and 5 for $M_U=1$ and 4, respectively.

Figure 4

Fidelity density between the quasi-exact simulation and the time-evolved l-USC state with $N_q=6$ and varied number of layers $M_U$. Inset: The entanglement entropies obtained within optimization of the l-USC circuit at $N_q=6$. (a) The integrable case $h=0$. (b) Nonintegrable case $h=J$.

Figure 5

(a) Maximum (over evolution time) error of approximating the exact environment by the $M_E$-layered environment throughout the full simulation shown in Fig. 3. Here we consider $h=g=J$. The black dashed line shows the environment approximation error at $M_E=N_q$. (b) Expectation value $\langle {\widehat{\sigma }}^z\left(t\right)\rangle$ obtained within the full algorithm of time evolution of l-USC with $M_U=1$ and layered environment with $M_E=N_q$. Left inset: The reachable time $t^{*}$ of simulations with the exact environment and the layered environment with $M_E=N_q$. The lines show linear fits. Right inset: The error in fidelity density $1-\mathcal{F}$ for different $N_q$.

Figure 6

(a) Expectation value of $|\langle {\widehat{\sigma }}^z\left(t\right)\rangle {|}^2$ obtained by various methods: quasi-exact (iTEBD), the presented algorithm with $N_q=2,M_U=M_E=1$ (l-USC), simulation of an ideal quantum device (simulator), and on the real hardware ibmq-jakarta (QPU) at $h/J=0, g/J=0.2$. (b) $h/J=0.05, g/J=0.8$.

Figure 7

(a) A typical l-USC circuit representation of a variational state. (b) Same with internal (not physical and not projected) indices denoted as ${\mathit{\alpha }}_k$. (c) The l-USC circuit rearranged to have a typical horizontal arrangement of uniform MPS. (d) The corresponding tensor network representation with uniform MPS $B^{\left[k\right]}$.

Figure 8

The total number of the gradient descent iterations $N_{\text{iter.}}\left(N_q\right)$ required to reach the time $t^{*}J$ using the procedure discussed in this Appendix.

Figure 9

(a) Infidelity density between the quasi-exact solution and the l-USC Ansatz at $M_U=1$ or uniform MPS at $\chi =2^{N_q-1}$ as a function of $tJ$ at $g/J=1.0, h/J=0.0$. (b) Entanglement entropy as a function of $tJ$. The solid line represents the quasi-exact solution. (c, d) Same results for the case $g/J=h/J=1.0$.

Figure 10

Comparison between the true infidelity density $1-\mathcal{F}$ computed using the exact solution (crosses) and using the cumulative estimation metric $\mathcal{M}\left(t\right)=1-{\prod }_{i<t}{\left|{\lambda }_i\right|}^2$. The simulations were performed at $g/J=1.0, h/J=0$ with an exact environment and state unitary with $M_U=1$.

Figure 11

The norm difference between the exact environment $|r_{\text{exact}}\rangle$ and the approximated $\left|r\right(M_E)\rangle$ environment at $M_E$ layers as a function of $tJ$. The ${10}^{-4}$ boundary is chosen in the main text as the threshold of accurate representation. The data are obtained within fitting of the exact environment emerging during the $N_q=6$ evolution of the l-USC Ansatz at $g/J=1.0$.

Figure 12

(a) Yang-Baxter equation (1). (b) Merging operation (2).

Figure 13

(a) Generalized Yang-Baxter transformation (3). To derive it, we repeatedly apply (1). (b) Absorbtion operation (4). After applying (3) to the green gates, the highlighted (orange) gate can be merged using (2). (c) Block reduction operation (5) by repeatedly applying (4).

Figure 14

Graphical proof of the theorem. The blue line represents the pivotal point that indicates which unitaries are being simplified.