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Register a new userQuantum annealing correction at finite temperature: ferromagnetic $p$-spin models
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The performance of open-system quantum annealing is adversely affected by thermal excitations out of the ground state. While the presence of energy gaps between the ground and excited states suppresses such excitations, error correction techniques are required to ensure full scalability of quantum annealing. Quantum annealing correction (QAC) is a method that aims to improve the performance of quantum annealers when control over only the problem (final) Hamiltonian is possible, along with decoding. Building on our earlier work [S. Matsuura et al., Phys. Rev. Lett. 116, 220501 (2016)], we study QAC using analytical tools of statistical physics by considering the effects of temperature and a transverse field on the penalty qubits in the ferromagnetic $p$-body infinite-range transverse-field Ising model. We analyze the effect of QAC on second ($p=2$) and first ($pgeq 3$) order phase transitions, and construct the phase diagram as a function of temperature and penalty strength. Our analysis reveals that for sufficiently low temperatures and in the absence of a transverse field on the penalty qubit, QAC breaks up a single, large free energy barrier into multiple smaller ones. We find theoretical evidence for an optimal penalty strength in the case of a transverse field on the penalty qubit, a feature observed in QAC experiments. Our results provide further compelling evidence that QAC provides an advantage over unencoded quantum annealing.
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To use quantum systems for technological applications we first need to preserve their coherence for macroscopic timescales, even at finite temperature. Quantum error correction has made it possible to actively correct errors that affect a quantum memory. An attractive scenario is the construction of passive storage of quantum information with minimal active support. Indeed, passive protection is the basis of robust and scalable classical technology, physically realized in the form of the transistor and the ferromagnetic hard disk. The discovery of an analogous quantum system is a challenging open problem, plagued with a variety of no-go theorems. Several approaches have been devised to overcome these theorems by taking advantage of their loopholes. Here we review the state-of-the-art developments in this field in an informative and pedagogical way. We give the main principles of self-correcting quantum memories and we analyze several milestone examples from the literature of two-, three- and higher-dimensional quantum memories.
The probability of success of quantum annealing can be improved significantly by pausing the annealer during its dynamics, exploiting thermal relaxation in a controlled fashion. In this paper, we investigate the effect of pausing the quantum annealing of the fully-connected ferromagnetic $ p $-spin model. We numerically show that (i) the optimal pausing point is 60% longer than the avoided crossing time for the analyzed instance, and (ii) at the optimal pausing point, we register a 45% improvement in the probability of success with respect to a quantum annealing with no pauses of the same duration. These results are in line with those observed experimentally for less connected models with the available quantum annealers. The observed improvement for the $ p $-spin model can be up to two orders of magnitude with respect to an isolated quantum dynamics of the same duration.
We investigate the quantum annealing of the ferromagnetic $ p $-spin model in a dissipative environment ($ p = 5 $ and $ p = 7 $). This model, in the large $ p $ limit, codifies the Grovers algorithm for searching in an unsorted database. The dissipative environment is described by a phonon bath in thermal equilibrium at finite temperature. The dynamics is studied in the framework of a Lindblad master equation for the reduced density matrix describing only the spins. Exploiting the symmetries of our model Hamiltonian, we can describe many spins and extrapolate expected trends for large $ N $, and $ p $. While at weak system bath coupling the dissipative environment has detrimental effects on the annealing results, we show that in the intermediate coupling regime, the phonon bath seems to speed up the annealing at low temperatures. This improvement in the performance is likely not due to thermal fluctuation but rather arises from a correlated spin-bath state and persists even at zero temperature. This result may pave the way to a new scenario in which, by appropriately engineering the system-bath coupling, one may optimize quantum annealing performances below either the purely quantum or classical limit.
We present a general error-correcting scheme for quantum annealing that allows for the encoding of a logical qubit into an arbitrarily large number of physical qubits. Given any Ising model optimization problem, the encoding replaces each logical qubit by a complete graph of degree $C$, representing the distance of the error-correcting code. A subsequent minor-embedding step then implements the encoding on the underlying hardware graph of the quantum annealer. We demonstrate experimentally that the performance of a D-Wave Two quantum annealing device improves as $C$ grows. We show that the performance improvement can be interpreted as arising from an effective increase in the energy scale of the problem Hamiltonian, or equivalently, an effective reduction in the temperature at which the device operates. The number $C$ thus allows us to control the amount of protection against thermal and control errors, and in particular, to trade qubits for a lower effective temperature that scales as $C^{-eta}$, with $eta leq 2$. This effective temperature reduction is an important step towards scalable quantum annealing.
Nested quantum annealing correction (NQAC) is an error correcting scheme for quantum annealing that allows for the encoding of a logical qubit into an arbitrarily large number of physical qubits. The encoding replaces each logical qubit by a complete graph of degree $C$. The nesting level $C$ represents the distance of the error-correcting code and controls the amount of protection against thermal and control errors. Theoretical mean-field analyses and empirical data obtained with a D-Wave Two quantum annealer (supporting up to $512$ qubits) showed that NQAC has the potential to achieve a scalable effective temperature reduction, $T_{rm eff} sim C^{-eta}$, with $eta leq 2$. We confirm that this scaling is preserved when NQAC is tested on a D-Wave 2000Q device (supporting up to $2048$ qubits). In addition, we show that NQAC can be also used in sampling problems to lower the effective temperature of a quantum annealer. Such effective temperature reduction is relevant for machine-learning applications. Since we demonstrate that NQAC achieves error correction via an effective reduction of the temperature of the quantum annealing device, our results address the problem of the temperature scaling law for quantum annealers, which requires the temperature of quantum annealers to be reduced as problems of larger sizes are attempted to be solved.
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