We then fully characterize the structural constraints imposed on quantum processes with finite Markov order, shedding light on a variety of memory effects that can arise through various examples. As measurements are inherently invasive in quantum mechanics, one has no choice but to define Markov order with respect to the interrogating instruments that are used to probe the process at hand: different memory effects are exhibited depending on how one addresses the system, in contrast to the standard classical setting. This is achieved by constructing a quantum Markov order condition, which naturally generalizes its classical counterpart for the quantification of finite-length memory effects. In this paper, using a novel framework for the characterization of quantum stochastic processes, we first solve the long standing question of unambiguously describing the memory length of a quantum processes. This issue can be overcome by separating the experimental interventions from the underlying process, enabling an unambiguous description of the process itself and accounting for all possible multi-time correlations for any choice of interrogating instruments. However, attempting to generalize this notion to quantum mechanics is problematic: observing realizations of a quantum process necessarily disturbs the state of the system, breaking an implicit, and crucial, assumption in the classical setting. This is well understood in classical theory, where a hierarchy of joint probability distributions completely characterizes the process at hand. Capturing the complete multi-time statistics that define a stochastic process lies at the heart of any proper treatment of memory effects. Understanding temporal processes and their correlations in time is of paramount importance for the development of near-term technologies that operate under realistic conditions. Suggest that progressively reducing noise level in qubits and gates is as important asĬontinuously integrating more qubits to realize scalable and reliable quantum computer. Is reduced O(\sqrt(n)) times below the accuracy threshold, arbitrarily accurate quantumĬomputation becomes feasible with acceptable scaling of the codeword size. However, if instead, per-operation qubit error probability in an n-qubits long codeword Incrementing the codeword size while retaining constant noise level per qubit operation. Noise correlation, one cannot guarantee arbitrary high computational accuracy simply by Likelihood of the occurring large numbers of qubits errors. Phase-flipped qubits, decays sub-exponentially in the size of the set and carries non-trivial In this noise model, the probability distribution over set of Model which permits collective coupling of all the codeword qubits to the same non. Qubit codes to, for instance, oscillators and rotors.We investigate the efficacy of topological quantum error-correction in correlated noise We systematically construct codesĬovariant with respect to general groups, obtaining natural generalizations of Unitary group, achieving good accuracy for large $d$ (using random codes) or We construct codes covariant with respect to the full logical Physical qubits per subsystem that is inversely proportional to the error Theorem: If a code admits a universal set of transversal gates and correctsĮrasure with fixed accuracy, then, for each logical qubit, we need a number of Representation theory, we prove an approximate version of the Eastin-Knill Of infidelity with $n$ or $d$ as the lower bound. We exhibit codes achieving approximately the same scaling This boundĪpproaches zero when the number of subsystems $n$ or the dimension $d$ of each $G$-covariant code with $G$ a continuous group, we derive a lower bound on theĮrror correction infidelity following erasure of a subsystem. Realized by performing transformations on the individual subsystems. Symmetry group $G$ if a $G$ transformation on the logical system can be $n$ physical subsystems, we say that the code is covariant with respect to a ![]() If a logical quantum system is encoded into Including many-body systems, metrology in the presence of noise, fault-tolerantĬomputation, and holographic quantum gravity. Albert, Grant Salton, Fernando Pastawski, Patrick Hayden, John Preskill Download PDF Abstract: Quantum error correction and symmetry arise in many areas of physics, Authors: Philippe Faist, Sepehr Nezami, Victor V.
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