Preparing 100-qubit symmetry-protected topological order on a digital quantum computer

TL;DR

The results establish a practical foundation for probing non-equilibrium quench dynamics of symmetry-protected topological systems in regimes that challenge classical computational methods and establishes current quantum computers as versatile platforms for large-scale studies of symmetry-protected quantum matter.

Abstract

Symmetry-protected topological (SPT) phases extend the Landau paradigm of quantum matter by admitting distinct symmetry-preserving phases that lack any local order parameter. Demonstrating these phases at scale on programmable quantum processors is a key milestone in using quantum hardware to probe emergent many-body phenomena, yet it is impeded by the circuit depth normally required to capture non-trivial entanglement. Here we use a tensor network based approximate quantum compiling (AQC) protocol to construct shallow quantum circuits (18-39 CNOT depth), which prepare 100-site ground states of the spin-1/2 bond-alternating Heisenberg chain in both SPT phases, to 97.9-99.0% fidelity. Upon executing the circuits on IBM quantum hardware, the resulting states exhibit all defining signatures of SPT order including non-local string order for strings of up to length 20, characteristic degeneracies in the entanglement spectrum and clear evidence of symmetry-protected edge modes. The simultaneous observation of these independent diagnostics establishes current quantum computers as versatile platforms for large-scale studies of symmetry-protected quantum matter. More broadly, our results establish a practical foundation for probing non-equilibrium quench dynamics of such systems in regimes that challenge classical computational methods.

Figures (5)

• Figure 1: a) (Top) A diagram representing the 100-site spin-1/2 bond-alternating Heisenberg chain, with alternating $J_0$, $J_1$ couplings. (Bottom) a schematic representation of the ground states for the positive $J_0$ and $J_1$ axes, consisting of simple products of singlet pairs: $1/\sqrt{2}(\ket{\uparrow\downarrow}-\ket{\downarrow\uparrow})$, plus decoupled edge spins in the odd-Haldane phase. b) The phase diagram of the model haghshenas2014symmetrywang2013topological. The model exhibits two phases: the odd-Haldane phase for $J_1>0$ and $J_1>J_0$ and the even-Haldane phase for $J_0>0$ and $J_0>J_1$, along with a trivial ferromagnetic phase for $J_0<0$ and $J_1<0$. The black crosses denote the four points in the phase diagram considered in this work, which we denote as $O_\frac{1}{2}$, $E_\frac{1}{2}$, $E_{-1}$, and $E_{-2}$.
• Figure 2: (a) A schematic of our workflow. We use to find the ground state of the Hamiltonian, Eq. \ref{['eq:bahc_hamiltonian']}. Then we employ , specifically using the AQC-Tensor algorithm with a brickwork ansatz, to approximately map the target to a shallow quantum circuit. Finally, we execute the circuit on quantum hardware. The coupling map shown is that of ibm_pittsburgh, with the blue qubits and connections signifying those used for the experiment shown in Fig. \ref{['fig:3_string_order']}, and the ends of the chain highlighted in red. (b) A sketch illustrating how the string order, Eq. \ref{['eq:string_order']}, is measured (right), and the tomography procedure for obtaining the entanglement spectrum (left). Here the $\lambda_i$'s are the sorted eigenvalues of the $l$-site reduced density matrix, $\hat{\rho}^{(l)}$, with $\lambda_1 \geq \lambda_2 \geq~...$
• Figure 3: (a) Odd (left) and even (right) string order parameters as a function of string length for the $E_{-2}$ ground state, prepared and measured using ibm_pittsburgh. Each data point is averaged over five sections of the chain, corresponding to $s=20,~30,~40,~50$, and 60. The blue (red) crosses show the expectation values with (without) . The error bars represent the standard error on the mean (red) and the uncertainty of the fit evaluated at a noise factor of zero (blue), after propagation for the average value using the relation for a sum. The black and green crosses show the corresponding values obtained classically from and from simulation of the compiled circuit, respectively. The black dashed line represents the maximum possible value of 1. (b) A schematic of how the $l=20$ even string order parameter is obtained by averaging over multiple sections of the chain. (Left) ibm_pittsburgh coupling map showing the qubits used in this experiment (blue) along with the footprints of the 20-local even string order observables (red) for $s=20$ and $60$. (Right) curves for the two observables. The red crosses show the measured values as a function of noise factor, and the blue curve shows the extrapolation.
• Figure 4: String order parameters and single-site magnetisation of the $O_\frac{1}{2}$ ground state, prepared and measured using ibm_pittsburgh. (a) Odd (left) and even (right) string order parameters as a function of string length. Each data point is averaged over five sections of the chain, corresponding to $s=20,~30,~40,~50$, and 60. The black dashed line represents the maximum possible value of 1. (b) Single-site magnetisation, $\braket{\hat{Z}_i}/2$, of the twenty sites closest to each end of the chain. The fluctuations at the edges correspond to the exponentially-localised spin-1/2 edge states. In both panels, the blue (red) crosses show the expectation values with (without) . The black and green crosses show the corresponding values obtained classically from and from simulation of the compiled circuit, respectively. The error bars represent the standard error on the mean (red) and the uncertainty of the fit evaluated at a noise factor of zero (blue). In panel (a), the uncertainties on the mean over the five sections are calculated using the relation for a sum.
• Figure 5: Entanglement spectra of the (a) $O_\frac{1}{2}$ and (b) $E_{-2}$ ground states for cuts of up to $l=6$ sites, executed on ibm_pittsburgh. The red (blue) bars show the eigenvalues of the reduced density matrix in descending order, $\lambda_1 \geq\lambda_2\geq\ldots$, obtained by cutting $J_0$ ($J_1$) bonds. The error bars are obtained via a bootstrapping procedure and represent the standard deviation over 1000 samples of the Pauli-string expectation values. See Methods Sec. \ref{['subsubsec:entanglement_spectrum']} for more details. The black crosses show the corresponding values obtained from .