Encoding a qubit in an oscillator

Quantum error-correcting codes are constructed that embed a finite-dimensional code space in the infinite-dimensional Hilbert space of a system described by continuous quantum variables. These codes exploit the noncommutative geometry of phase space to protect against errors that shift the values of the canonical variables q and p. In the setting of quantum optics, fault-tolerant universal quantum computation can be executed on the protected code subspace using linear optical operations, squeezing, homodyne detection, and photon counting; however, nonlinear mode coupling is required for the preparation of the encoded states. Finite-dimensional versions of these codes can be constructed that protect encoded quantum information against shifts in the amplitude or phase of a d-state system. Continuous-variable codes can be invoked to establish lower bounds on the quantum capacity of Gaussian quantum channels.Comment: 22 pages, 8 figures, REVTeX, title change (qudit -> qubit) requested by Phys. Rev. A, minor correction

Demonstrating Quantum Error Correction that Extends the Lifetime of Quantum Information

The remarkable discovery of Quantum Error Correction (QEC), which can overcome the errors experienced by a bit of quantum information (qubit), was a critical advance that gives hope for eventually realizing practical quantum computers. In principle, a system that implements QEC can actually pass a "break-even" point and preserve quantum information for longer than the lifetime of its constituent parts. Reaching the break-even point, however, has thus far remained an outstanding and challenging goal. Several previous works have demonstrated elements of QEC in NMR, ions, nitrogen vacancy (NV) centers, photons, and superconducting transmons. However, these works primarily illustrate the signatures or scaling properties of QEC codes rather than test the capacity of the system to extend the lifetime of quantum information over time. Here we demonstrate a QEC system that reaches the break-even point by suppressing the natural errors due to energy loss for a qubit logically encoded in superpositions of coherent states, or cat states of a superconducting resonator. Moreover, the experiment implements a full QEC protocol by using real-time feedback to encode, monitor naturally occurring errors, decode, and correct. As measured by full process tomography, the enhanced lifetime of the encoded information is 320 microseconds without any post-selection. This is 20 times greater than that of the system's transmon, over twice as long as an uncorrected logical encoding, and 10% longer than the highest quality element of the system (the resonator's 0, 1 Fock states). Our results illustrate the power of novel, hardware efficient qubit encodings over traditional QEC schemes. Furthermore, they advance the field of experimental error correction from confirming the basic concepts to exploring the metrics that drive system performance and the challenges in implementing a fault-tolerant system

Fault-tolerant quantum computation

The discovery of quantum error correction has greatly improved the long-term prospects for quantum computing technology. Encoded quantum information can be protected from errors that arise due to uncontrolled interactions with the environment, or due to imperfect implementations of quantum logical operations. Recovery from errors can work effectively even if occasional mistakes occur during the recovery procedure. Furthermore, encoded quantum information can be processed without serious propagation of errors. In principle, an arbitrarily long quantum computation can be performed reliably, provided that the average probability of error per gate is less than a certain critical value, the accuracy threshold. It may be possible to incorporate intrinsic fault tolerance into the design of quantum computing hardware, perhaps by invoking topological Aharonov-Bohm interactions to process quantum information.Comment: 58 pages with 7 PostScript figures, LaTeX, uses sprocl.sty and psfig, to appear in "Introduction to Quantum Computation," edited by H.-K. Lo, S. Popescu, and T. P. Spille

Fault-Tolerant Measurement-Based Quantum Computing with Continuous-Variable Cluster States

A long-standing open question about Gaussian continuous-variable cluster states is whether they enable fault-tolerant measurement-based quantum computation. The answer is yes. Initial squeezing in the cluster above a threshold value of 20.5 dB ensures that errors from finite squeezing acting on encoded qubits are below the fault-tolerance threshold of known qubit-based error-correcting codes. By concatenating with one of these codes and using ancilla-based error correction, fault-tolerant measurement-based quantum computation of theoretically indefinite length is possible with finitely squeezed cluster states.Comment: (v3) consistent with published version, more accessible for general audience; (v2) condensed presentation, added references on GKP state generation and a comparison of currently achievable squeezing to the threshold; (v1) 13 pages, a few figure

Adiabatic Quantum Simulators

In his famous 1981 talk, Feynman proposed that unlike classical computers, which would presumably experience an exponential slowdown when simulating quantum phenomena, a universal quantum simulator would not. An ideal quantum simulator would be controllable, and built using existing technology. In some cases, moving away from gate-model-based implementations of quantum computing may offer a more feasible solution for particular experimental implementations. Here we consider an adiabatic quantum simulator which simulates the ground state properties of sparse Hamiltonians consisting of one- and two-local interaction terms, using sparse Hamiltonians with at most three-local interactions. Properties of such Hamiltonians can be well approximated with Hamiltonians containing only two-local terms. The register holding the simulated ground state is brought adiabatically into interaction with a probe qubit, followed by a single diabatic gate operation on the probe which then undergoes free evolution until measured. This allows one to recover e.g. the ground state energy of the Hamiltonian being simulated. Given a ground state, this scheme can be used to verify the QMA-complete problem LOCAL HAMILTONIAN, and is therefore likely more powerful than classical computing.Comment: 9 pages, 1 figur

Internal Consistency of Fault-Tolerant Quantum Error Correction in Light of Rigorous Derivations of the Quantum Markovian Limit

We critically examine the internal consistency of a set of minimal assumptions entering the theory of fault-tolerant quantum error correction for Markovian noise. These assumptions are: fast gates, a constant supply of fresh and cold ancillas, and a Markovian bath. We point out that these assumptions may not be mutually consistent in light of rigorous formulations of the Markovian approximation. Namely, Markovian dynamics requires either the singular coupling limit (high temperature), or the weak coupling limit (weak system-bath interaction). The former is incompatible with the assumption of a constant and fresh supply of cold ancillas, while the latter is inconsistent with fast gates. We discuss ways to resolve these inconsistencies. As part of our discussion we derive, in the weak coupling limit, a new master equation for a system subject to periodic driving.Comment: 19 pages. v2: Significantly expanded version. New title. Includes a debate section in response to comments on the previous version, many of which appeared here http://dabacon.org/pontiff/?p=959 and here http://dabacon.org/pontiff/?p=1028. Contains a new derivation of the Markovian master equation with periodic drivin