Classically optimized Hamiltonian simulation

Hamiltonian simulation is a promising application for quantum computers to achieve a quantum advantage. We present classical algorithms based on tensor network methods to optimize quantum circuits for this task. We show that, compared to Trotter product formulas, the classically optimized circuits can be orders of magnitude more accurate and significantly extend the total simulation time.

Figure 1

Classically optimized Hamiltonian simulation (thick lines) vs Trotter product formulas (thin lines) for a brickwall circuit of depth (a) 1, (b) 2, and (c) 3. The approximation error, defined in Sec. 2, of the approximate evolution operator $U_{\text{approx}}$ with respect to the exact one $exp(-itH)$ is shown as a function of time $t$. We consider $H=2{\sum }_{k=1}^7{Z}_k{Z}_{k+1}+{\sum }_{k=1}^8{X}_k+{\sum }_{k=1}^8{Z}_k$. For the Trotter results $U_{\text{approx}}$ is a Trotter product formula of first (dotted), second (dashed), and fourth (solid) order. For the classically optimized results $U_{\text{approx}}=U\left(\mathit{\theta }\right)$ is the circuit with parameters $\mathit{\theta }$ after two optimization procedures. The first procedure minimizes the approximation error to the exact operator $exp(-itH)$ (solid); the second procedure cuts the total time into $S=100$ (dashed), 200 (dash-dotted), and 300 (dotted) slices of equal time $\tau =1/S$ and then optimizes using a first-order Taylor approximation of $exp(-i\tau H)$ and $S$ iterations thereof. We observe that by increasing $S,$ the results of the Taylor approach converge to the ones obtained via $exp(-itH)$. Additionally we see that the classically optimized two- and three-layer circuits are two orders of magnitude more accurate than the Trotter formulas. Further details are in Sec. 3.

Figure 2

(a) Brickwall ansatz for $n=5$ qubits. (b) Parameterization in terms of the gates $R_x\left(\theta \right)=exp(-i\theta X/2), R_z\left(\theta \right)=exp(-i\theta Z/2),$ and $U_{zz}\left(\theta \right)=exp(-i\theta Z\otimes Z/2),$ where $\theta$ denotes the variational parameter.

Figure 3

Approximation error as a function of time for classically optimized brickwall circuits of $L$ layers and $n$ qubits.

Figure 4

(a) Approximation error and (b) phase error of the classically optimized circuit $U\left(\mathit{\theta }\right)$ achieved for various qubit counts $n$. We cut the total time into $S=200$ slices of equal time $\tau =1/S$ and then optimize using L-BFGS, a first-order Taylor approximation of $exp(-i\tau H),$ and $S$ iterations thereof. We consider the Hamiltonian in Eq. (7) and a brickwall circuit of two layers $(L=2)$. The insets show the approximation (a) and phase (b) errors normalized by the number of qubits $n$. In both cases the collapse of the data onto a single curve indicates a linear scaling of the errors with qubit number.

Figure 5

Basic examples of tensor network diagrams. (a) A vector corresponds to a node with one leg which represents the index of the vector. (b) A matrix has two legs for the row and column indices of the matrix. (c) A third-order tensor has three legs for three indices. (d) A fourth-order tensor has four legs for four indices. (e) The product of a matrix and a vector gives another vector. The summation in the matrix-vector product over the connecting indices between the matrix, and the vector is represented by the connecting edge between the nodes. (f) The tensor contraction of a third-order tensor with a matrix over one index gives a third-order tensor. (g) The tensor contraction of two third-order tensors over one index gives a fourth-order tensor.

Figure 6

The two-qubit gate of our ansatz is composed of the two-qubit gate $U_{zz}$ and the one-qubit gates $R_x$ and $R_z$. The one-qubit gates are represented as matrices in the tensor notation, while the two-qubit $U_{zz}$ gate is decomposed into a pair of third-order tensors and a bond matrix which connects them according to Eq. (A1). These tensors are contracted to give a pair of third-order tensors, which represents the entire two-qubit gate of our ansatz.

Figure 7

Decomposition of one brickwall layer of our ansatz into a MPO for $n=5$ qubits. Each two-qubit gate is first decomposed into a pair of third-order tensors as described in Fig. 6. Then we contract adjacent tensors via the horizontally connecting edges to form a MPO.

Figure 8

A pair of MPOs is represented by a single new MPO. We contract the MPOs such that the bond dimension of the new MPO is the product of those of the contracted MPOs, as shown in the central figure. We find an approximate MPO representation of smaller bond dimension using the compression techniques in [33].

Figure 9

Trace over a product of three MPOs as required for the evaluation of Eq. (1). We first contract the three tensors in the topmost dashed rectangle into one large tensor represented by an oval node with three indices. Then we contract this large tensor with the three adjacent tensors one after another. The procedure is repeated $O\left(n\right)$ times until the final contraction, with the bottom-most three tensors giving the desired scalar number representing the trace over the MPO product.

Figure 10

Spectral-norm distance as a function of time for classically optimized brickwall circuits of $L$ layers, $n$ qubits, and for the Hamiltonian of Eq. (7) with parameters $(J,g,h)=(2.0,1.0,1.0)$. The classical optimization was performed using the exact method outlined in Sec. 3. A constant factor increase in the spectral-norm distance is observed as $n$ increases, while the error scalings (slopes of the linear parts of the curves) are approximately independent of the system size.

Figure 11

Approximation error of the classically optimized circuit $U\left(\mathit{\theta }\right)$ with respect to the exact time evolution operator $exp(-itH)$ achieved using various optimization algorithms. The variational parameters $\mathit{\theta }$ are initialized to zero, and a single iteration corresponds to the update of all variational parameters in the circuit once. We consider the Hamiltonian of Eq. (7) with parameters $(J,g,h)=(2.0,1.0,1.0)$, time $t=0.05, n=8$ qubits, and a brickwall circuit of two layers.

Figure 12

Mean (top) and median (bottom) absolute gradients of the first step $(s=1)$ of the sequential optimization procedure using random initialization. Data are for system sizes of $n$ qubits and a range of Taylor time steps $\tau$.

Figure 13

Mean (top) and median (bottom) absolute gradients of the first step $(s=1)$ of the sequential optimization procedure using identity initialization. Data are for system sizes of $n$ qubits and a range of Taylor time steps $\tau$.

Figure 14

Average absolute gradients (mean and median) as a function of time for systems of $n=10$ and $n=60$ qubits. We compute the gradient of the cost function before the first classical optimization step (e.g., L-BFGS) at each slice $s$. Data are presented as a function of time where $t=\tau s$.

Figure 15

Classically optimized Hamiltonian simulation (thick lines) vs Trotter product formulas (thin lines) for a brickwall circuit of depth (a) 1, (b) 2, and (c) 3. The approximation error of the approximate evolution operator $U_{\text{approx}}$ with respect to the exact one $exp(-itH)$ is shown as a function of time $t$. We consider $H=2{\sum }_{k=1}^4{Z}_k{Z}_{k+1}+{\sum }_{k=1}^5{X}_k+{\sum }_{k=1}^5{Z}_k$. For the Trotter results $U_{\text{approx}}$ is a Trotter product formula of first (dotted), second (dashed), and fourth (solid) order. For the classically optimized results, $U_{\text{approx}}=U\left(\mathit{\theta }\right)$ is the circuit with parameters $\mathit{\theta }$ after either the exact (thick solid) or Taylor-based (thick dashed) optimization procedures as outlined in Sec. 3.

Figure 16

Time-evolved $n=12$ qubit state infidelity averaged over 50 random initial bitstring product states plotted as a function of time for (a) circuits of six layers and (b) circuits of twelve layers. We consider the Hamiltonian of Eq. (7) with parameters $(J,g,h)=(1.0,1.0,1.0)$. The time evolution circuits approximating $exp(-itH)$ are constructed by appending multiple second-order Trotter circuits (thin dashed line) or multiple classically optimized circuits (thick lines) as explained in the text. The solid horizontal lines approximate the total infidelity due to noisy two-qubit gates for various two-qubit gate error rates $p$. The error bars on the data are negligibly small compared to the scale of the figure and are therefore omitted for clarity.