Programmable Quantum Annealing Architectures with Ising Quantum Wires

Quantum annealing aims at solving optimization problems efficiently by preparing the ground state of an Ising spin-Hamiltonian quantum mechanically. A prerequisite of building a quantum annealer is the implementation of programmable long-range two-, three-, or multispin Ising interactions. We discuss an architecture, where the required spin interactions are implemented via two-port or in general multiport quantum Ising wires connecting the spins of interest. This quantum annealing architecture of spins connected by Ising quantum wires can be realized by exploiting the three-dimensional (3D) character of atomic platforms, including atoms in optical lattices and Rydberg tweezer arrays. The realization only requires engineering on-site terms and two-body interactions between nearest neighboring qubits. The locally coupled spin model on a 3D cubic lattice is sufficient to effectively produce arbitrary all-to-all coupled Ising Hamiltonians. We illustrate the approach for few-spin devices solving Max-Cut and prime factorization problems, and discuss the potential scaling to large atom-based systems.

Popular Summary

Quantum annealing aims at solving search and optimization problems benefiting from a quantum advantage. During the last years, quantum annealing has received growing interest in view of far-reaching scientific and commercial applications. Atomic systems provide a controllable platform with unique scalability in the effort to implement quantum annealers. A key challenge is the physical realization of individually programmable long-range interactions between the spins (or qubits). In this work, we propose to engineer long-range interactions via locally coupled Ising ferromagnetic quantum wires, where a quantum annealer with all-to-all connectivity is mapped to a local architecture on a regular 3D cubic lattice.

We show that a locally coupled spin model on a 3D cubic lattice, with only on-site fields and nearest-neighbor two-body interactions, is sufficient to effectively produce arbitrary all-to-all coupled Ising Hamiltonians. The underlying idea of a quantum wire mediating long-range two-body Ising interactions can be further generalized to $m$-port quantum wires providing many-body interactions on complex geometries. The resultant local quantum annealing architecture can be realized by utilizing the 3D character of atomic platforms, including atoms in optical lattices and Rydberg tweezer arrays. We illustrate with few-spin simulations the potential of the present architecture to solve problems such as Max-Cut and prime factorization.

While the present work focuses on programmable quantum annealers with all-to-all connectivity, a demonstration of the basic building blocks such as $m$-port quantum wires and quantum-wire-induced interactions already provides interesting opportunities for quantum simulation of exotic spin physics.

Figure 1

Illustration of the three-dimensional cubic architecture for quantum annealing. (a) The Ising ferromagnetic quantum-wire-induced interaction, with a ferromagnetic quantum wire coupling to qubits ${\hat{\sigma }}_i^z$ and ${\hat{\sigma }}_{i^{\prime }}^z$ at the two ends. The auxiliary spins in the wire are denoted as ${\hat{\tau }}_{j=0,1,\dots ,M-1}^z$ (the number of ancilla $M$ is five in this example), which are ferromagnetically coupled by $J<0$, with an Ising Hamiltonian. The coupling between ${\hat{\tau }}_0^z$ (${\hat{\tau }}_{M-1}^z$) and ${\hat{\sigma }}_i^z$ (${\hat{\sigma }}_{i^{\prime }}^z$) is $J_L$ ($J_R$). The ground state energy of the quantum wire with the two distant qubits restricted to an eigenstate subspace of ${\hat{\sigma }}_i^z$ and ${\hat{\sigma }}_{i^{\prime }}^z$ is listed in the table, which determines the induced effective interaction between the two qubits. (b) The $Y$-junction connector for engineering a three-body interaction. With three quantum wires forming a $Y$ junction that couples ${\hat{\sigma }}_{A,B,C}^z$ through ${\hat{\tau }}_o^z$ [Eq. (10)], an effective three-body interaction is mediated (see the main text). (c) The geometry of the three-dimensional cubic quantum annealing architecture. The blue spheres represent the $N$ duplicated copies of logical qubits. The cubes represent the introduced ancilla. The light and dark grey links correspond to ferromagnetic couplings with fixed strengths $J$ and $J_d$. The red links correspond to the programmable Ising couplings [Eq. (8a)]. All couplings in this quantum annealing architecture are local and at most between two neighboring sites, and can be embedded in a regular cubic lattice. In this example we set $N=5$ for illustration.

Figure 2

The probability distribution of $l_{\max }$ in Ising spin glass ground states with random couplings. We simulate the Ising spin glass models with all-to-all random couplings with $4$, $8$, $12$, $16$, and $20$ vertices. The random couplings are drawn from $[-1,1]$ according to a uniform distribution. We obtain the ground state and calculate the value of $l_{\max }$ (see the main text). The statistics are taken over samples of ${10}^5$ random problem instances. It is evident that the probability distribution $p(l_{\max })$ exhibits an exponential decay at large $l_{\max }$.

Figure 3

The error rate ($p_{\mathrm{err}}$) in the ferromagnetic quantum wire at finite temperature ($T$). We choose a series of wire lengths, $M=10,40,70,100,130,160$. The gray dash line indicates a threshold probability of $p_{\mathrm{th}}=1/8$. The inset shows the corresponding temperature threshold $T_{\mathrm{th}}$ with different $M$, calculated from the exact expression in Eq. (14) and the approximation in Eq. (15).

Figure 4

Demonstration of the 3D cubic architecture for quantum annealing applied to Max-Cut problems. For the graphs shown at the top right of (a) and (e), we encode the Max-Cut problem of the graph in a spin glass model. The four logical qubits correspond to the four vertices in the graph. The corresponding 3D encoding architecture is shown in (a) and (e) with the unnecessary layers removed. The respective numerical performances are shown in (b) and (f), taking the local quantum annealing Hamiltonian in Eq. (8a). Plots of the fidelity and the average numbers of defects in the duplication qubits, ${\mathrm{ND}}_{\mathrm{dup}}$, and in the connecting quantum wires, ${\mathrm{ND}}_{\mathrm{qw}}$, against the total adiabatic time ${\tau }_{\mathrm{ad}}$ (see the main text). The insets show the monotonic decay behaviors of ${\mathrm{ND}}_{\mathrm{dup}}$ and ${\mathrm{ND}}_{\mathrm{qw}}$ at large ${\tau }_{\mathrm{ad}}$. For both cases, the quantum annealer reaches a fidelity of $50\mathrm{\%}$ around ${\tau }_{\mathrm{ad}}=20$. The defect numbers ${\mathrm{ND}}_{\mathrm{dup}}$ and ${\mathrm{ND}}_{\mathrm{qw}}$ drop to below 1% at about ${\tau }_{\mathrm{ad}}=30$, and decrease rapidly beyond that. The corresponding instantaneous eigenstate energy spectra for the lowest forty states are shown in (d) and (h). For comparison, the energy spectra with the direct nonlocal Hamiltonian in Eq. (2) are shown in (c) and (g). The energy spectra are all shifted with respect to the instantaneous ground state energy, and the dashed lines thus represent the ground state levels in (c), (d), (g), (h). The coupling in the Ising formulation of the Max-Cut problems is taken as an energy unit (see the main text). We choose the parameters $J_d=-1.1D_{\max }$ and $J=-1.5$.

Figure 5

Demonstration of the 3D cubic architecture for quantum annealing applied to prime factorization. A prime factorization of $15$ is investigated. This problem is mapped to a spin glass model of four vertices with all-to-all couplings. (a) The 3D cubic quantum annealing architecture with $26$ qubits. (b) Plots of the fidelity and the average numbers of defects in the duplication qubits, ${\mathrm{ND}}_{\mathrm{dup}}$, and in the quantum wire connectors, ${\mathrm{ND}}_{\mathrm{qw}}$, against the total adiabatic time ${\tau }_{\mathrm{ad}}$. The defect numbers ${\mathrm{ND}}_{\mathrm{dup}}$ and ${\mathrm{ND}}_{\mathrm{qw}}$ are rescaled for cosmetic reasons. Here we choose the parameters $J_d=-1.1D_{\max }{K}_{\max }$ and $J=-1.5K_{\max }$. The defect excitations become negligible for an adiabatic time ${\tau }_{\mathrm{ad}}>20$. (d) The instantaneous eigenstate energy spectra of the local quantum annealing Hamiltonian in Eq. (8a). We calculate the lowest twenty states. (c) The energy spectra for quantum annealing with the direct nonlocal three-body Hamiltonian [see Eq. (20)]. The energy spectra are shifted with respect to the instantaneous ground state energy, and the dashed lines thus represent the ground state levels in (c) and (d). The coupling strength of the three-body interaction is taken as an energy unit.

Figure 6

Illustration of Rydberg $p$-wave building blocks for implementation of the 3D cubic quantum annealing architecture. The atoms at the sites of spheres and cubes are dressed by the Rydberg states ${|n{}^2{P}_{3/2},m=3/2⟩}_i$ and ${|n{}^2{P}_{3/2},m=3/2⟩}_k$, respectively, with the subscripts $i$ and $k$ denoting the quantization axes. Solid (dashed) lines indicate the presence (absence) of couplings between the two Rydberg dressed atoms. For the angular dependence of the Rydberg $p$-wave interaction [Eq. (21)], in the horizontal $i$-$k$ plane the duplicated logical qubits (ancilla) only couple to each other along the $k$ ($i$) direction as shown in (a),(b). Along the vertical $h$ direction, both the duplicated logical qubits and the ancilla interact with their neighbors, as shown in (a)–(c). The red link representing a programmable Ising coupling is realized by the mixed interaction between the two different Rydberg states. This coupling orientation corresponds to $(\theta =\pi /2,\varphi =\pi /2)$ in the angular dependence of the mixed interaction.