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arXiv:1904.12028 · Physical Review A

Arbitrary-time error suppression for Markovian adiabatic quantum computing using stabilizer subspace codes

Daniel A. Lidar

Published Aug 20, 2019• 10 citations
arXivHTMLPDF

Abstract

Adiabatic quantum computing (AQC) can be protected against thermal excitations via an encoding into error-detecting codes, supplemented with an energy penalty formed from a sum of commuting Hamiltonian terms. Earlier work showed that it is possible to suppress the initial thermally induced excitation out of the encoded ground state, in the case of local Markovian environments, by using an energy-penalty strength that grows only logarithmically in system size, at a fixed temperature. The question of whether this result applies beyond the initial time was left open. Here we answer this in the affirmative. We show that thermal excitations out of the encoded ground state can be suppressed at arbitrary times under the additional assumption that the total evolution time is polynomial in the system size. Thus, computational problems that can be solved efficiently by using AQC in a closed system setting can still be solved efficiently subject to coupling to a thermal environment. Our construction uses stabilizer subspace codes, which require at least 4-local interactions to achieve this result.

DOI
10.1103/physreva.100.022326

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Adiabatic quantum computing (AQC) can be protected against thermal excitations via an encoding into error-detecting codes, supplemented with an energy penalty formed from a sum of commuting Hamiltonian terms. Earlier work showed that it is possible to suppress the initial thermally induced excitation out of the encoded ground state, in the case of local Markovian environments, by using an energy-penalty strength that grows only logarithmically in system size, at a fixed temperature. The question of whether this result applies beyond the initial time was left open. Here we answer this in the affirmative. We show that thermal excitations out of the encoded ground state can be suppressed at arbitrary times under the additional assumption that the total evolution time is polynomial in the system size. Thus, computational problems that can be solved efficiently by using AQC in a closed system setting can still be solved efficiently subject to coupling to a thermal environment. Our construction uses stabilizer subspace codes, which require at least 4-local interactions to achieve this result.
Hamiltonian quantum computing, such as the adiabatic and holonomic models, can be protected against decoherence using an encoding into stabilizer subspace codes for error detection and the addition of energy penalty terms. This method has been widely studied since it was first introduced by Jordan, Farhi, and Shor (JFS) in the context of adiabatic quantum computing. Here, we extend the original result to general Markovian environments, not necessarily in Lindblad form. We show that the main conclusion of the original JFS study holds under these general circumstances: Assuming a physically reasonable bath model, it is possible to suppress the initial decay out of the encoded ground state with an energy penalty strength that grows only logarithmically in the system size, at a fixed temperature.
We consider the use of quantum error-detecting codes, together with energy penalties against leaving the code space, as a method for suppressing environmentally induced errors in Hamiltonian-based quantum computation. This method was introduced in Jordan et al. [Phys. Rev. A 74, 052322 (2006)] in the context of quantum adiabatic computation, but we consider it more generally. Specifically, we consider a computational Hamiltonian, which has been encoded using the logical qubits of a single-qubit error-detecting code, coupled to an environment of qubits by interaction terms that act one-locally on the system. Additional energy penalty terms penalize states outside of the code space. We prove that in the limit of infinitely large penalties, one-local errors are completely suppressed, and we derive some bounds for the finite penalty case. Our proof technique involves exact integration of the Schrodinger equation, making no use of master equations or their assumptions. We perform long time numerical simulations on a small (one logical qubit) computational system coupled to an environment and the results suggest that the energy penalty method achieves even greater protection than our bounds indicate.
One of the main methods for protecting quantum information against decoherence is to encode information in the ground subspace (or the low energy sector) of a Hamiltonian with a large energy gap which penalizes errors from environment. The protecting Hamiltonian is chosen such that its degenerate ground subspace is an error detecting code for the errors caused by the interaction with the environment. We consider environments with arbitrary number of local sites, e.g. spins, whose interactions among themselves are local and bounded. Then, assuming the system is interacting with a finite number of sites in the environment, we prove that, up to second order with respect to the coupling constant, decoherence and relaxation are suppressed by a factor which grows exponentially fast with the ratio of energy penalty to the norm of local interactions in the environment. The state may, however, still evolve unitarily inside the code subspace due to the Lamb shift effect. In the context of adiabatic quantum computation, this means that the evolution inside the code subspace is effectively governed by a renormalized Hamiltonian. The result is derived from first principles, without use of master equations or their assumptions, and holds even in the infinite temperature limit. We also prove that unbounded or non-local interactions in the environment at sites far from the system do not considerably modify the exponential suppression. Our main technical tool is a new bound on the decay of power spectral density at high frequencies for local observables and many-body Hamiltonians with bounded and local interactions in a neighborhood around the support of the observable.
While adiabatic quantum computing (AQC) has some robustness to noise and decoherence, it is widely believed that encoding, error suppression and error correction will be required to scale AQC to large problem sizes. Previous works have established at least two different techniques for error suppression in AQC. In this paper we derive a model for describing the dynamics of encoded AQC and show that previous constructions for error suppression can be unified with this dynamical model. In addition, the model clarifies the mechanisms of error suppression and allows the identification of its weaknesses. In the second half of the paper, we utilize our description of non-equilibrium dynamics in encoded AQC to construct methods for error correction in AQC by cooling local degrees of freedom (qubits). While this is shown to be possible in principle, we also identify the key challenge to this approach: the requirement of high-weight Hamiltonians. Finally, we use our dynamical model to perform a simplified thermal stability analysis of concatenated-stabilizer-code encoded many-body systems for AQC or quantum memories. This work is a companion paper to 'Error suppression and error correction in adiabatic quantum computation: techniques and challenges (2013 Phys. Rev. X 3 041013)', which provides a quantum information perspective on the techniques and limitations of error suppression and correction in AQC. In this paper we couch the same results within a dynamical framework, which allows for a detailed analysis of the non-equilibrium dynamics of error suppression and correction in encoded AQC.
We present general conditions for quantum error suppression for Hamiltonian-based quantum computation using subsystem codes. This involves encoding the Hamiltonian performing the computation using an error detecting subsystem code and the addition of a penalty term that commutes with the encoded Hamiltonian. The scheme is general and includes the stabilizer formalism of both subspace and subsystem codes as special cases. We derive performance bounds and show that complete error suppression results in the large penalty limit. To illustrate the power of subsystem-based error suppression, we introduce fully two-local constructions for protection against local errors of the swap gate of adiabatic gate teleportation and the Ising chain in a transverse field.
Achieving noise resilience is an outstanding challenge in Hamiltonian-based quantum computation. To this end, energy-gap protection provides a promising approach, where the desired quantum dynamics are encoded into the ground space of a penalty Hamiltonian that suppresses unwanted noise processes. However, existing approaches either explicitly require high-weight penalty terms that are not directly accessible in current hardware, or utilize non-commuting 222-local Hamiltonians, which typically leads to an exponentially small energy gap. In this work, we provide a general recipe for designing error-resilient Hamiltonian simulations, making use of an excited encoding subspace stabilized by solely 222-local commuting Hamiltonians. Our results thus overcome a no-go theorem previously derived for ground-space encoding that prevents noise suppression schemes with such Hamiltonians. Importantly, our method is scalable as it only requires penalty terms that scale polynomially with system size. To illustrate the utility of our approach, we further apply this method to a variety of 111- and 222-dimensional many-body spin models, potentially extending the duration of high-fidelity simulation by orders of magnitude in current hardware.