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arXiv:1410.5487 · Physical Review Letters

Quantum Error Suppression with Commuting Hamiltonians: Two Local is Too Local

Iman Marvian, Daniel A. Lidar

Published Dec 31, 2014• 20 citations
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Abstract

We consider error suppression schemes in which quantum information is encoded into the ground subspace of a Hamiltonian comprising a sum of commuting terms. Since such Hamiltonians are gapped, they are considered natural candidates for protection of quantum information and topological or adiabatic quantum computation. However, we prove that they cannot be used to this end in the two-local case. By making the favorable assumption that the gap is infinite, we show that single-site perturbations can generate a degeneracy splitting in the ground subspace of this type of Hamiltonian which is of the same order as the magnitude of the perturbation, and is independent of the number of interacting sites and their Hilbert space dimensions, just as in the absence of the protecting Hamiltonian. This splitting results in decoherence of the ground subspace, and we demonstrate that for natural noise models the coherence time is proportional to the inverse of the degeneracy splitting. Our proof involves a new version of the no-hiding theorem which shows that quantum information cannot be approximately hidden in the correlations between two quantum systems. The main reason that two-local commuting Hamiltonians cannot be used for quantum error suppression is that their ground subspaces have only short-range (two-body) entanglement.

DOI
10.1103/physrevlett.113.260504

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We consider error suppression schemes in which quantum information is encoded into the ground subspace of a Hamiltonian comprising a sum of commuting terms. Since such Hamiltonians are gapped, they are considered natural candidates for protection of quantum information and topological or adiabatic quantum computation. However, we prove that they cannot be used to this end in the two-local case. By making the favorable assumption that the gap is infinite, we show that single-site perturbations can generate a degeneracy splitting in the ground subspace of this type of Hamiltonian which is of the same order as the magnitude of the perturbation, and is independent of the number of interacting sites and their Hilbert space dimensions, just as in the absence of the protecting Hamiltonian. This splitting results in decoherence of the ground subspace, and we demonstrate that for natural noise models the coherence time is proportional to the inverse of the degeneracy splitting. Our proof involves a new version of the no-hiding theorem which shows that quantum information cannot be approximately hidden in the correlations between two quantum systems. The main reason that two-local commuting Hamiltonians cannot be used for quantum error suppression is that their ground subspaces have only short-range (two-body) entanglement.
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.
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.
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.
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.
We present a new scheme to perform noise resilient universal adiabatic quantum computation using two-body interactions. To achieve this, we introduce a new family of error detecting subsystem codes whose gauge generators and a set of their logical operators - capable of encoding universal Hamiltonian computations - can be implemented using two-body interactions. Logical operators of the code are used to encode any given computational Hamiltonian, and the gauge operators are used to construct a penalty Hamiltonian whose ground subspace is protected against local-errors. In contrast to previous approaches, the constructed penalty Hamiltonian does not necessarily commute with the encoded computational Hamiltonians, but for our construction, the undesirable effect of the penalty Hamiltonian on the computation can be compensated by a simple modification of the implemented Hamiltonians. We also investigate whether a similar scheme can be constructed by encoding the computational Hamiltonian using only bare-logical operators of subsystem codes, to guarantee that computational Hamiltonian commutes with the penalty Hamiltonian. We prove a no-go theorem showing that restricting to two-body interactions and using any general CSS-type subsystem codes, such a construction cannot encode systems beyond an Ising chain in a transverse field. We point out that such a chain is universal for Hamiltonian-based quantum computation, but it is not universal for ground-state quantum computation.
Abstract Incorporating protection against quantum errors into adiabatic quantum computing (AQC) is an important task due to the inevitable presence of decoherence. Here, we investigate an error-protected encoding of the AQC Hamiltonian, where qubit ensembles are used in place of qubits. Our Hamiltonian only involves total spin operators of the ensembles, offering a simpler route towards error-corrected quantum computing. Our scheme is particularly suited to neutral atomic gases where it is possible to realize large ensemble sizes and produce ensemble-ensemble entanglement. We identify a critical ensemble size Nc where the nature of the first excited state becomes a single particle perturbation of the ground state, and the gap energy is predictable by mean-field theory. For ensemble sizes larger than Nc, the ground state becomes protected due to the presence of logically equivalent states and the AQC performance improves with N, as long as the decoherence rate is sufficiently low.