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arXiv:2411.07117

Analytically Exact Quantum Simulation of N-Body Interactions via Untunable Decentralized Hamiltonians for Implementing the Toric Code and Its Modifications

Haochen Zhao, Florian Mintert

Published Nov 11, 2024• 0 citations
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Abstract

We propose a new quantum simulation method for simulating N-body interactions, which are tensor products of N Pauli operators, in an analytically exact manner. This method iteratively attaches many two-body interactions on one two-body interaction to simulate an N-body interaction. Those controlled two-body interactions can be untunable and act only on neighboring spins. The strength difference between controlled and target Hamiltonians is normally not more than one order of magnitude. This simulation is theoretically error-free, and errors due to experimental imperfections are ignorable. A major obstacle to simulating the toric code model and modified toric codes used in topological quantum computation is to simulate N-body interactions. We employ the new quantum simulation method to solve this issue and thus simulate the toric code model and its modifications.

DOI
10.48550/arxiv.2411.07117

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We propose a new quantum simulation method for simulating N-body interactions, which are tensor products of N Pauli operators, in an analytically exact manner. This method iteratively attaches many two-body interactions on one two-body interaction to simulate an N-body interaction. Those controlled two-body interactions can be untunable and act only on neighboring spins. The strength difference between controlled and target Hamiltonians is normally not more than one order of magnitude. This simulation is theoretically error-free, and errors due to experimental imperfections are ignorable. A major obstacle to simulating the toric code model and modified toric codes used in topological quantum computation is to simulate N-body interactions. We employ the new quantum simulation method to solve this issue and thus simulate the toric code model and its modifications.
Many-body entangled systems, in particular topologically ordered spin systems proposed as resources for quantum information processing tasks, often involve highly nonlocal interaction terms. While one may approximate such systems through two-body interactions perturbatively, these approaches have a number of drawbacks in practice. In this Letter, we propose a scheme to simulate many-body spin Hamiltonians with two-body Hamiltonians nonperturbatively. Unlike previous approaches, our Hamiltonians are not only exactly solvable with exact ground state degeneracy, but also support completely localized quasiparticle excitations, which are ideal for quantum information processing tasks. Our construction is limited to simulating the toric code and quantum double models, but generalizations to other nonlocal spin Hamiltonians may be possible.
Topological order is now being established as a central criterion for characterizing and classifying ground states of condensed matter systems and complements categorizations based on symmetries. Fractional quantum Hall systems and quantum spin liquids are receiving substantial interest because of their intriguing quantum correlations, their exotic excitations, and prospects for protecting stored quantum information against errors. Here, we show that the Hamiltonian of the central model of this class of systems, the toric code, can be directly implemented as an analog quantum simulator in lattices of superconducting circuits. The four-body interactions, which lie at its heart, are in our concept realized via superconducting quantum interference devices (SQUIDs) that are driven by a suitably oscillating flux bias. All physical qubits and coupling SQUIDs can be individually controlled with high precision. Topologically ordered states can be prepared via an adiabatic ramp of the stabilizer interactions. Strings of qubit operators, including the stabilizers and correlations along noncontractible loops, can be read out via a capacitive coupling to read-out resonators. Moreover, the available single-qubit operations allow to create and propagate elementary excitations of the toric code and to verify their fractional statistics. The architecture we propose allows to implement a large variety of many-body interactions and thus provides a versatile analog quantum simulator for topological order and lattice gauge theories.
Quantum error correction allows for faulty quantum systems to behave in an effectively error-free manner. One important class of techniques for quantum error correction is the class of quantum subsystem codes, which are relevant both to active quantum error-correcting schemes as well as to the design of self-correcting quantum memories. Previous approaches for investigating these codes have focused on applying theoretical analysis to look for interesting codes and to investigate their properties. In this paper we present an alternative approach that uses computational analysis to accomplish the same goals. Specifically, we present an algorithm that computes the optimal quantum subsystem code that can be implemented given an arbitrary set of measurement operators that are tensor products of Pauli operators. We then demonstrate the utility of this algorithm by performing a systematic investigation of the quantum subsystem codes that exist in the setting where the interactions are limited to two-body interactions between neighbors on lattices derived from the convex uniform tilings of the plane.
We present here a procedure to obtain the Hamiltonians of the toric code and Kitaev quantum double models as the low-energy limits of entirely two-body Hamiltonians. Our construction makes use of a new type of perturbation gadget based on error-detecting subsystem codes. The procedure is motivated by a projected entangled pair states (PEPS) description of the target models, and reproduces the target models' behavior using only couplings that are natural in terms of the original Hamiltonians. This allows our construction to capture the symmetries of the target models.
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In a recent experiment, Barreiro et al (2011 Nature 470 486) demonstrated the fundamental building blocks of an open-system quantum simulator with trapped ions. Using up to five ions, dynamics were realized by sequences that combined single- and multi-qubit entangling gate operations with optical pumping. This enabled the implementation of both coherent many-body dynamics and dissipative processes by controlling the coupling of the system to an artificial, suitably tailored environment. This engineering was illustrated by the dissipative preparation of entangled two- and four-qubit states, the simulation of coherent four-body spin interactions and the quantum non-demolition measurement of a multi-qubit stabilizer operator. In this paper, we present the theoretical framework of this gate-based ('digital') simulation approach for open-system dynamics with trapped ions. In addition, we discuss how within this simulation approach, minimal instances of spin models of interest in the context of topological quantum computing and condensed matter physics can be realized in state-of-the-art linear ion-trap quantum computing architectures. We outline concrete simulation schemes for Kitaev's toric code Hamiltonian and a recently suggested color code model. The presented simulation protocols can be adapted to scalable and two-dimensional ion-trap architectures, which are currently under development.