Holographic quantum algorithms for simulating correlated spin systems

We present a suite of “holographic” quantum algorithms for efficient ground-state preparation and dynamical evolution of correlated spin systems, which require far fewer qubits than the number of spins being simulated. The algorithms exploit the equivalence between matrix-product states (MPS) and quantum channels, along with partial measurement and qubit reuse, in order to simulate a $D$-dimensional spin system using only a $(D-1)$-dimensional subset of qubits along with an ancillary qubit register whose size scales logarithmically in the amount of entanglement present in the simulated state. Ground states can either be directly prepared from a known MPS representation or obtained via a holographic variational quantum eigensolver (holoVQE). Dynamics of MPS under local Hamiltonians for time $t$ can also be simulated with an additional (multiplicative) $\mathrm{poly}\left(t\right)$ overhead in qubit resources. These techniques open the door to efficient quantum simulation of MPS with exponentially large bond dimension, including ground states of two- and three-dimensional systems, or thermalizing dynamics with rapid entanglement growth. As a demonstration of the potential resource savings, we implement a holoVQE simulation of the antiferromagnetic Heisenberg chain on a trapped-ion quantum computer, achieving within $10\left(3\right)\%$ of the exact ground-state energy of an infinite chain using only a pair of qubits.

Figure 1

Graphical description of an MPS. (a) An individual tensor, (b) the MPS wave function as a contraction over such tensors, and (c) the contraction of a tensor network to compute a correlation function.

Figure 2

MPS as a quantum circuit. (a) An MPS, assumed to be in right canonical form. The right canonical condition [Eq. (2)] guarantees that $V$ is an isometry from ${\mathbb{C}}^{\chi }\to {\mathbb{C}}^{\chi }\otimes {\mathbb{C}}^{\mathcal{Q}}$, and it can therefore be embedded in a unitary $U_V$ acting on ${\mathbb{C}}^{\chi }\otimes {\mathbb{C}}^{\mathcal{Q}}$ but restricted to a fixed input of the physical qubit (denoted with an open circle), as in panel (b). The unitary evolution in panel (b) is equivalent (up to simply rearranging the lines) to the circuit diagram in panel (c).

Figure 3

A circuit implementing holographic state preparation and correlation function measurement of an MPS. In this simple example, we use one system qubit (enabling simulation of an infinite 1D spin-$1/2$ chain) and $n_{\mathrm{b}}$ bond qubits for an MPS with bond dimension $\chi =2^{n_{\mathrm{b}}}$. Measurements are implied to be in the eigenbasis of the operators ${\widehat{O}}_n$ and ${\widehat{O}}_{n+r}$. Averaging over the outcome of this circuit (with the two measurement results multiplied together) produces the correlation function $\langle {\widehat{O}}_n{\widehat{O}}_{n+r}\rangle$.

Figure 4

Top: Energy per site of the $XXZ$ chain in the thermodynamic limit. The dashed line is mean-field theory, the solid line is the exact energy obtained from Bethe ansatz, and the points are from simulated holoVQE using a single bond qubit (error bars represent $1\sigma$ sampling uncertainties). Bottom: Fractional energy error $[\delta E=|(E-E_{\mathrm{exact}})/E_{\mathrm{exact}}\left|\right]$ of holoVQE (pink points with error bars) and mean-field theory (black points), showing that the addition of even a single bond qubit can drastically improve the ground-state energy estimate.

Figure 5

Experimental implementation of holoVQE on a Honeywell trapped-ion quantum computer. (a) Schematic of the ion trap; for this experiment we used two neighboring gate zones, each loaded with two data qubits and two sympathetic cooling ions (not shown). (b) Decomposition of ${\mathcal{G}}_{\theta }$ into two $\mathrm{c}Z$ gates. (c) The circuits used for holoVQE involve one physical qubit and one bond qubit, though we utilize the two gate zones to parallelize the data taking. (d) Representative data from holoVQE.

Figure 6

(a) Circuits used for holographic MPS representation of the TFIM ground state. Each arbitrary SU(4) is decomposed into three native two-qubit gates and eight single-qubit gates, with a total of 15 real variational parameters (as in Ref. [37]). (b) Energy obtained using “star” circuits with an increasing number of bond qubits. The agreement with exact MPS energy minimization at bond dimension $\chi =2^{n_{\mathrm{b}}}$ is excellent, although we were not able to obtain reliable energies from this ansatz for $n_{\mathrm{b}}\ge 3$.

Figure 7

Correlation functions of the transverse (a) and longitudinal (b) spin directions in the 1D TFIM. Black lines are exact from fermionization, while the points are from holoVQE (blue points for $n_{\mathrm{b}}=1$ and red points for $n_{\mathrm{b}}=2$). In panel (b), the $\chi =8$ curve is shown as well (black dashed line), though this has been obtained from direct MPS optimization as the numerical minimization for an $n_{\mathrm{b}}=3$ star circuit was inconclusive.

Figure 8

Holographic time evolution of a matrix product state for nearest neighbor interactions ($k=2$). The circuit can be evaluated by first executing all gates in the past causal cone (gray shaded region) of the qubits exiting the top left corner of the circuit. The remaining slices can be executed in order by (a) resetting the qubits exiting the top of the previous slice, (b) using the reset qubits to extend the MPS in space, and (c) applying all time-evolution gates in the current slice.

Figure 9

Circuit for holographic time evolution of an MPS, obtained from Fig. 8 by attaching wires exiting the top of the circuit to those at the bottom wherever the respective exiting qubits are reset and reused at the bottom of the circuit. The space direction wraps diagonally around the cylinder indefinitely (the site of the physical system at position $x$ is labeled $s_x$), and the circumference of the cylinder, which determines the required number of qubits, is determined by the evolution time. A time-ordered correlation function $\langle \widehat{O}(x_1,t_1)\widehat{O}(x_2,t_2)\rangle$ is obtained by measuring the corresponding operators at appropriate places in the circuit, as shown here (green circles representing measurements) for the example $x_1=2$, $x_2=2r+1$.