Session V15: Quantum Error Correction and Control

The structure of preserved information in quantum processes

We present a general operational framework for characterizing the types of information that can be preserved by a quantum process. We demonstrate that \emph{information preserving structures} (IPS) -- encompassing noiseless subsystems, decoherence-free subspaces, pointer bases, and error-correcting codes -- are isometric to fixed points of unital quantum processes. This implies that every IPS is a matrix algebra. A structure theorem for fixed points of an arbitrary process further provides a simple and efficient algorithm for finding all noiseless and unitarily noiseless IPS for any quantum process. This framework can be extended to study the structure of approximately preserved information.   

Decoherence-free subspaces and incoherently generated coherences

A decoherence-free subspace (DFS) is a collection of states that is immune to the dominant noise effects created by the environment. DFS is usually studied for states involving two or more particles and is considered a prominent candidate for quantum memory and quantum information processing. We present rigorous criteria for the existence of DFS in finite-dimensional systems coupled to the Markovian reservoirs. This allows us to identify a new special class of decoherence free states that relies on rather counterintuitive phenomenon, which we call an ``incoherent generation of coherences.'' We provide examples of physical systems that support such states.   

Methods for Producing Decoherence-free States and Noiseless Subsystems Using Photonic Qutrits

We outline a proposal for a method of preparing a single logically encoded two-state system (qubit) that is immune to collective noise acting on the Hilbert space of the particles supporting it. The logical qubit is comprised of three photonic 3-state systems (qutrits) and is generated by the process of spontaneous parametric down-conversion. The states are constructed using linear optical elements along with three down-conversion sources, and are deemed successful by the simultaneous detection of six events. We also show how to select a maximally entangled state of two qutrits by similar methods. For this maximally entangled state we describe conditions for the state to be decoherence-free which do not correspond to collective errors, but which have a precisely defined relationship between them.   

Bounded-strength dynamical control of qubit coherence based on Eulerian cycles

Decoherence and faulty controls are two of the primary obstacles to realize scalable quantum information processing. Here, we investigate dynamical decoupling (DD) techniques for dynamical control and decoherence suppression in the limit of low-power faulty control, using the approach of Eulerian DD introduced in L. Viola and E. Knill, Phys. Rev. Lett. 90, 037901 (2003). By focusing on the illustrative case of single-qubit DD, we identify scenarios where naive transcriptions of bang-bang sequences with finite pulses are outperformed by the Eulerian method -- both in terms of DD fidelity and robustness against systematic errors. Results on Eulerian decoherence control in solid-state qubit devices are presented.   

Model-independent dynamical decoupling to combat dephasing decoherence

We present a remarkable finding that a recently [1] discovered series of pulse sequences, designed to optimally restore coherence to a qubit in the spin-boson model of decoherence, is in fact completely model-independent and generically valid for arbitrary dephasing Hamiltonians given sufficiently short delay times between pulses [2]. The series is optimal in that fidelity is maximized for a given number of applied pulses. This is true for sufficiently short delay times because the series, with each pulse, cancels successive orders of a time expansion for the decay of qubit fidelity. Surprisingly, this property is independent of the model of the bath that induces dephasing-type decoherence. For this to be true, a linearly growing set of ``unknowns'' (the delay times) simultaneously satisfy an exponentially growing set of non-linear equations. This is an unexpected and miraculous property of nature and mathematics. [1] G. S. Uhrig, Phys. Rev. Lett. 98, 100504 (2007). [2] B. Lee, W. M. Witzel, S. Das Sarma, arXiv:0710.1416.   

Encoding One Logical Qubit Into Six Physical Qubits

We discuss several methods to protect one qubit against single- qubit errors by encoding it into six physical qubits. We first present a degenerate six-qubit quantum error-correcting code. We explicitly provide the stabilizer generators, encoding circuit, codewords, logical Pauli operators, and logical CNOT operator for this code. We then prove that a six-qubit code cannot simultaneously possess a Calderbank-Shor-Steane stabilizer and correct arbitrary single-qubit errors. We finally construct a six-qubit non-degenerate entanglement-assisted quantum error- correcting code that uses one bit of entanglement shared between the sender and the receiver. We discuss the advantages and disadvantages for each of our six-qubit quantum error-correcting codes.   

Second-order self-refocusing pulse shapes for arbitrary rotation angles

We construct several families of high-precision 1st- and 2nd-order self-refocusing pulse shapes for rotation angles $\alpha=0^\circ$, $10^\circ$, $\ldots$, $360^\circ$. To characterize their performance, we show that for an arbitrarily-coupled qubit driven by a general one-dimensional symmetric pulse shape, in addition to the net rotation angle, the second-order average Hamiltonian is defined by three parameters. Our 1st- and 2nd-order self-refocusing pulses respectively have one or two of these equal to zero, which makes them useful as a drop-in replacement for hard pulses. We illustrate this by analyzing several commonly-used composite pulses in terms of the average Hamiltonian theory. The results are in an excellent agreement with numerical simulations.   

Encoded Dynamical Recoupling with Shaped Pulses

Encoded Dynamical Recoupling is a passive error correction techique which can be used to enhance the performance of a quantum error correction code (QECC) against low-frequency component of the thermal bath. The elements of the stabilizer group are used in the decoupling cycle which makes the encoded logic operations fault-tolerant. We studied the effectiveness of this techique both analytically and numerically for several three- and five-qubit codes, with decoupling sequences utilizing either Gaussian or self-refocusing pulse shapes. When logic pulses are intercalated between the decoupling cycles, the technique may be very effective in cancelling constant perturbation terms, but its performance is much weaker against a time-dependent perturbation simulated as a classical correlated noise. The decoupling accuracy can be substantially improved if logic is applied slowly and concurrently with the decoupling, so that a certain adiabaticity condition is satisfied.   

Generating Novel Spin Echoes Using the Internal Structure of Strong Pi-Pulses

Conventionally, strong pulses used to control coherent evolution are approximated as instantaneous, perfect rotations. However, recent experiments have shown a surprising departure from the conventional theory in standard multipulse NMR experiments using strong pi-pulses. Using our understanding of the role that the finite time duration of any real pulse plays in these effects, we design and experimentally demonstrate new classes of spin echoes.   

A New Approach to Spin Coherence Control: Extreme Line-narrowing and MRI of Solids

The non-zero duration of strong pulses has been shown to have surprisingly large effects in important NMR experiments. The Hamiltonian terms arising from the internal structure of strong pulses provide us with a new technique of spin coherence control. Using this technique, we design and demonstrate new approaches to line-narrowing and magnetic resonance imaging of solids.   

Robustness of operator quantum error correction against imperfect initialization

It is known that perfect unitary evolution inside decoherence-free subspaces and subsystems (DFSs) requires more restrictive conditions if one allows imperfect initialization of the state. It was believed that these conditions are necessary if DFSs are to be able to protect imperfectly encoded states from subsequent errors. By a similar argument, general operator quantum error-correcting (OQEC) codes would also require more restrictive error-correction conditions for the case of imperfect initialization. In this study, we examine this requirement by looking at the errors on the encoded state. In order to quantitatively analyze the errors in an OQEC code, we introduce a measure of the fidelity between the encoded information in two states for the case of subsystem encoding. In contrast to what was previously believed, we obtain that more restrictive conditions are not necessary neither for DFSs nor for general OQEC codes. This is because the effective noise that can arise inside the code as a result of imperfect initialization is such that it can only increase the fidelity of an imperfectly encoded state with a perfectly encoded one. Some implications of this result in the context of fault-tolerant information processing are discussed.   

Quantum Convolutional Coding with Entanglement Assistance

We have recently developed quantum convolutional coding techniques for both entanglement distillation and quantum error correction. These techniques assume that the two parties participating in the communication protocols possess prior shared entanglement. Using these methods, we can import arbitrary classical binary or quaternary convolutional codes for use in quantum coding, with no requirement that these codes be self-orthogonal. Moreover, high-performance classical convolutional codes lead to high-performance quantum convolutional codes. We explicitly show how a convolutional entanglement distillation protocol operates, and how to encode and decode a stream of quantum information in an entanglement-assisted quantum convolutional code.   

``Slow'' and ``Fast'' Gate Limits In Resilient Quantum Computation

In the study of resilient quantum computation, there are two common approaches to proving the threshold theorem: one either uses a stochastic error model or uses the operator norm to bound the effects of the noise. In many cases, the underlying microscopic Hamiltonian is hidden due to the rapidly growing complexity of the problem. In particular, the microscopic interacting Hamiltonian in the interaction picture depends on the quantum code and its implementation. Nevertheless, there are two possible ways to keep the discussion code independent. The first situation is to imagine very fast gates acting on the system. The second is to derive an upper bound on the effects of correlations by deriving an effective model. In this talk we discuss these two limits, focusing on how to derive the effective model for ``slow gates''.   

Optimal Control of Large Spin-Atomic Systems with Coherent Electromagnetic Fields

Cold atomic systems provide an excellent testing ground for quantum control protocols due to the isolation of these systems from their environment and the availability of high precision fields from the ``quantum optics toolbox''. In this talk, we look at controlling the sixteen dimensional ground state hyperfine manifold of $^{133}$Cs through microwaves and rf-magnetic fields. These controls allow for essentially coherent manipulation of a system that is large enough to exhibit non-trivial dynamics. In particular, we analyze the controllability of this system under different combinations of applied electromagnetic fields. Also, we present a scheme for performing state preparation, which is the mapping a fiducial state to an arbitrary target state, and show simulations that examine the performance of these state preparation protocols.~~~~~   

Towards a Quantum Predictive Control Mathematical Formulation

Quantum Feedback Control presents an interdisciplinary research work between Control Theory and Quantum Physics. A novel mathematical formulation is presented for Quantum Predictive Control (QPC) algorithm, based on Receding Horizon Control philosophy at the quantum level. The application of Heisenberg Uncertainty Principle, to quantify the uncertainty in position and momentum, in a real-time control framework is introduced. An estimate of the Sampling Time at quantum level is demonstrated, applying to a case of study for a double well quantum system using the QPC algorithm.