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39:26 University of Southern California (USC) English 2011

#### Polar codes for classical, private, and quantum communication

Channel polarization is a phenomenon in which a particular recursive encoding induces a set of synthesized channels from many instances of a memoryless channel, such that a fraction of the synthesized channels become near perfect for data transmission and the other fraction become near useless for this task. The channel polarization effect then leads to a simple scheme for data transmission: send the information bits through the perfect channels and "frozen" bits through the useless ones. One contribution of the present work is to leverage several known results from the quantum information literature to demonstrate that the channel polarization effect takes hold for channels with classical inputs and quantum outputs. We construct linear polar codes based on this effect, and we also demonstrate that a quantum successive cancellation decoder works well, by exploiting Sen's recent "non-commutative union bound" that holds for a sequence of projectors applied to a quantum state. In addition, consider that Mahdavifar and Vardy have recently exploited the channel polarization phenomenon to construct codes that achieve the symmetric private capacity for private data transmission over a degraded wiretap channel. We build on their work and demonstrate how to construct quantum wiretap polar codes that achieve the symmetric private capacity of a degraded quantum wiretap channel with a classical eavesdropper. Due to the Schumacher-Westmoreland correspondence between quantum privacy and quantum coherence, we can construct quantum polar codes by operating these quantum wiretap polar codes in superposition. Our scheme achieves the symmetric coherent information rate for quantum channels that are degradable with a classical environment. This condition on the environment may seem restrictive, but we show that many quantum channels satisfy this criterion, including amplitude damping channels, photon-detected jump channels, dephasing channels, erasure channels, and cloning channels. Our quantum polar coding scheme has the desirable properties of being channel-adapted and symmetric capacity-achieving.
• Published: 2011
• Publisher: University of Southern California (USC)
• Language: English
34:29 University of Southern California (USC) English 2011

#### Quantum computers still work with 25% of their qubits missing

I will describe recent results from an ongoing project which examines the robustness of Kitaev's surface codes, and related FTQC schemes (due to Raussendorf and coworkers) to loss errors. The key insight is that, in a topologically ordered system, the quantum information is encoded in delocalized degrees of freedom that can be "deformed" to avoid missing physical qubits. This allows one to relate error correction and fault tolerance thresholds to percolation thresholds. Furthermore, stabilizer operators can be deformed in a similar way, which means that surface codes retain their robustness to arbitrary types of error, even when significant numbers of qubits are lost. We present numerical evidence, utilizing these insights, to show that: (1) the surface code can tolerate up to 50 percent loss errors, and (2) Raussendorf's FTQC scheme can tolerate up 25 percent loss errors. The numerics indicate both schemes retain good performance when loss and computational errors are simultaneously present. Finally we will describe extensions to other error models, in particular the case where logic gates can fail but in a heralded manner.
• Published: 2011
• Publisher: University of Southern California (USC)
• Language: English
39:42 University of Southern California (USC) English 2011

#### On the energy landscape of 3D spin Hamiltonians with topological order

We explore feasibility of a quantum self-correcting memory based on 3D spin Hamiltonians with topological quantum order in which thermal diffusion of topological defects is suppressed by macroscopic energy barriers. To this end we characterize the energy landscape of stabilizer code Hamiltonians with local bounded-strength interactions which have a topologically ordered ground state but do not have string-like logical operators. We prove that any sequence of local errors mapping a ground state of such Hamiltonian to an orthogonal ground state must cross an energy barrier growing at least as a logarithm of the lattice size. Our bound on the energy barrier is shown to be tight up to a constant factor for one particular 3D spin Hamiltonian.
• Published: 2011
• Publisher: University of Southern California (USC)
• Language: English
25:42 University of Southern California (USC) English 2011

#### Fault-tolerant quantum computing with color codes

We present and analyze protocols for fault-tolerant quantum computing using color codes. To process these codes, no qubit movement is necessary; nearest-neighbor gates in two spatial dimensions suffices. Our focus is on the color codes defined by the 4.8.8 semiregular lattice, as they provide the best error protection per physical qubit among color codes. We present circuit-level schemes for extracting the error syndrome of these codes fault-tolerantly. We further present an integer-program-based decoding algorithm for identifying the most likely error given the (possibly faulty) syndrome. We simulated our syndrome extraction and decoding algorithms against three physically-motivated noise models using Monte Carlo methods, and used the simulations to estimate the corresponding accuracy thresholds for fault-tolerant quantum error correction. We also used a self-avoiding walk analysis to lower-bound the accuracy threshold for two of these noise models. We present two methods for fault-tolerantly computing with these codes. In the first, many of the operations are transversal and therefore spatially local if two-dimensional arrays of qubits are stacked atop each other. In the second, code deformation techniques are used so that all quantum processing is spatially local in just two dimensions. In both cases, the accuracy threshold for computation is comparable to that for error correction. Our analysis demonstrates that color codes perform slightly better than Kitaev's surface codes when circuit details are ignored. When these details are considered, we estimate that color codes achieve a threshold of 0.082(3)\%, which is higher than the threshold of 1.3 \times 10^ achieved by concatenated coding schemes restricted to nearest-neighbor gates in two dimensions [Spedalieri and Roychowdhury, Quant.\ Inf.\ Comp.\ \textbf, 666 (2009)] but lower than the threshold of 0.75\% to 1.1\% reported for the Kitaev codes subject to the same restrictions [Raussendorf and Harrington, Phys.\ Rev.\ Lett.\ \textbf, 190504 (2007); Wang \etal, Phys. Rev. A \textbf, 020302(R) (2011)]. Finally, because the behavior of our decoder's performance for two of the noise models we consider maps onto an order-disorder phase transition in the three-body random-bond Ising model in 2D and the corresponding random-plaquette gauge model in 3D, our results also answer the Nishimori conjecture for these models in the negative: the statistical-mechanical classical spin systems associated to the 4.8.8 color codes are counterintuitively more ordered at positive temperature than at zero temperature.
• Published: 2011
• Publisher: University of Southern California (USC)
• Language: English
24:22 University of Southern California (USC) English 2011

#### Long range failure-tolerant entanglement distribution

We introduce a protocol to distribute entanglement between remote parties. Our protocol is based on a chain of repeater stations, and exploits topological encoding to tolerate very high levels of defect and error. The repeater stations may employ probabilistic entanglement operations which usually faill; ours is the first protocol to explicitly allow for technologies of this kind. Given a error rate between stations in excess of 10%, arbitrarily long range high fidelity entanglement distribution is possible even if the heralded failure rate within the stations is as high as 99%, providing that unheralded errors are low (order 0.01%).
• Published: 2011
• Publisher: University of Southern California (USC)
• Language: English
09:55 University of Southern California (USC) English 2011

#### QEC11 - Opening statement

• Published: 2011
• Publisher: University of Southern California (USC)
• Language: English
26:28 University of Southern California (USC) English 2011

#### Performance requirements of a quantum computer using surface code error correction

We study the various overhead costs associated with translating an abstract quantum algorithm to a practical implementation in fault-tolerant, error-corrected quantum hardware. The processes required for quantum error correction can be expensive in terms of quantum resources, and we consider the collective demands of the error correction circuits, distillation of ancilla states, and composition of arbitrary logical gates. To provide a concrete demonstration, we study a quantum computer architecture using surface code error correction, and we examine Shor's algorithm and simulation of quantum chemistry in first-quantized form as typical quantum algorithms for a large-scale quantum computer. As a consequence of this investigation, we can show that practical quantum computers executing these algorithms will require quantum hardware with physical gate operation times of less than 1 microsecond, if the calculation is to complete within 30 days for problems too difficult for existing classical processors.
• Published: 2011
• Publisher: University of Southern California (USC)
• Language: English
21:44 University of Southern California (USC) English 2011

#### Dynamical decoupling of encoded information

It is widely accepted that a quantum computer will need some form of protection against decoherence, and to date the only tool capable of providing fault-tolerance is quantum error correction (QEC). Here we study the interplay of Dynamical decoupling (DD), an open-loop non-fault-tolerant capable decoherence suppression method, and Quantum error correction codes. Several schemes exist where DD is used to improve gate fidelities, such as Dynamically Protected Gates or (Concatenated) Dynamically Corrected Gates. Several authors have combined DD and QEC with different degrees of success, showing improved gate fidelity but at the cost of, in cases strong, extra locality constraints on the error model or a sequence size which is exponential in the number of physical qubits (n). In this work we introduce a dynamical sequences that use as pulses elements of the stabilizer and the normalizer of the QEC code (SXDD sequences). We show how this idea can be adapted to the leading DD schemes: Concatenated DD and Nested Uhrig DD. With this we avoid the need for further locality constraints and obtain shorter sequences while still obtaining fidelity improvements. Moreover, the fact that the SXDD pulses can be chosen to be bitwise allows a natural integration with fault-tolerant methods and the direct porting of several DD schemes for fidelity-enhanced gates into the SXDD scenario.
• Published: 2011
• Publisher: University of Southern California (USC)
• Language: English
19:51 University of Southern California (USC) English 2011

#### Protecting quantum gates from control noise

External controls are necessary to enact quantum logic operations, and the inevitable control noise will result in gate errors in a realistic quantum circuit. We investigate the robustness of quantum gates to the random noise in an optimal control field, by utilizing properties of the quantum control landscape that relates the physical objective (in the present case, the quantum gate fidelity) to the applied controls. An approximate result obtained for the statistical expectation value of the gate fidelity in the weak noise regime is shown to be in excellent agreement with direct Monte Carlo sampling over noise process realizations for fidelity values relevant for practical quantum information processing. Using this approximate result, we demonstrate that maximizing the robustness to additive/multiplicative white noise is equivalent to minimizing the total control time/fluence. Also, a genetic optimization algorithm is used to identify controls with improved robustness to a colored noise characterized by its autocorrelation function.
• Published: 2011
• Publisher: University of Southern California (USC)
• Language: English
18:21 University of Southern California (USC) English 2011

#### Dynamical decoupling noise spectroscopy

Decoherence is one of the most important obstacles that must be overcome in quantum information processing. It depends on the qubit-environment coupling strength, but also on the spectral composition of the noise generated by the environment. If the spectral density is known, fighting the effect of decoherence can be made more effective. Applying sequences of inversion pulses to the qubit system, we generate effective filter functions that probe the environmental spectral density. Comparing different pulse sequences, we recover the complete spectral density function and distinguish different contributions to the overall decoherence.
• Published: 2011
• Publisher: University of Southern California (USC)
• Language: English
41:43 University of Southern California (USC) English 2011

#### Quantum Information Processing and Quantum Error Correction with Trapped Ca+ Ion

Trapped strings of cold ions provide an ideal system for quantum information processing. The quantum information can be stored in individual ions and these qubits can be individually prepared; the corresponding quantum states can be manipulated and measured with nearly 100% detection efficiency. With a small ion‐trap quantum computer based on up to fourteen trapped Ca+ ions as qubits we have generated genuine quantum states in a preprogrammed way. In particular, using high fidelity global and local quantum gate operations, with GHZ states of up to fourteen ions we have investigated noise and error sources during quantum information processing. Decoherence of multi‐qubit GHZ states was measured and is quantitatively described a noise model. Using optimized sequences of high fidelity gate operations enabled the implementation of repetitive quantum error correction. Future applications towards analog and digital quantum simulations will be indicated and briefly discussed.
• Published: 2011
• Publisher: University of Southern California (USC)
• Language: English
42:02 University of Southern California (USC) English 2011

#### Bound on quantum computation time: Quantum error correction in a critical environment

We obtain an upper bound on the time available for quantum computation for a given quantum computer and decohering environment with quantum error correction implemented. First, we derive an explicit quantum evolution operator for the logical qubits and show that it has the same form as that for the physical qubits but with a reduced coupling strength to the environment. Using this evolution operator, we find the trace distance between the real and ideal states of the logical qubits in two cases. For a super-Ohmic bath, the trace distance saturates, while for Ohmic or sub-Ohmic baths, there is a finite time before the trace distance exceeds a value set by the user.
• Published: 2011
• Publisher: University of Southern California (USC)
• Language: English
33:58 University of Southern California (USC) English 2011

#### Bulk fault-tolerant quantum information processing with boundary addressability

A globally controlled architecture for a quantum computer is highly appealing as one does not then require the technological capability of addressing each and every qubit within the device. Although several fully global controlled designs for a universal quantum computer have been proposed in the literature - and although it has been proven that a fully global fault tolerant (FT) quantum error correction scheme is possible in principle, no actual scheme for the latter has yet been advanced. In this work we go part way towards the latter and present a fault-tolerant semi-global control strategy for universal quantum computers. We show that an N-dimensional array of qubits where only (N − 1)-dimensional addressing resolution is available is compatible with FT universal quantum computation. What is more, we show that measurements and individual control of qubits are required only at the boundaries of the FT computer. Our model alleviates the heavy physical conditions on current qubit candidates imposed by addressability requirements and represents an option for improving their scalability.
• Published: 2011
• Publisher: University of Southern California (USC)
• Language: English
22:40 University of Southern California (USC) English 2011

#### Stabilizer quantum codes via the CWS framework

Codeword stabilized (CWS) construction defines a quantum code by combining a classical binary code with some underlying graph state. In general, CWS codes are non-additive but become additive stabilizer codes if derived from a linear binary code. Generic CWS codes typically require complex error correction; however, we show that the CWS framework is an efficient tool for constructing good stabilizer codes with simple decoding. We start by proving the lower Gilbert-Varshamov bound on the parameters of an additive CWS code which can be obtained from a given graph. We also show that cyclic additive CWS codes belong to a previously overlooked family of single-generator cyclic stabilizer codes; these codes are derived from a circulant graph and a cyclic binary code. Finally, we present several families of simple stabilizer codes with relatively good parameters, including a family of the smallest toric-like cyclic CWS codes which have length, dimension, and distance as follows: [[t2+(t+1)2,1,2t+1]], t=1,2, ...
• Published: 2011
• Publisher: University of Southern California (USC)
• Language: English
15:00 University of Southern California (USC) English 2011

#### A Class of Quantum Double Subsystem Codes

We introduce a family of codes which are a generalisation of the Bacon-Shor code using the quantum double models introduced by Kitaev. In particular, we show that such codes are possible for non-Abelian quantum double models, as well as three-dimensional quantum double models. Our codes possess a structure which generalises the gauge subsystem of the Bacon-Shor code to non-Abelian models. This means that we retain the ability to infer error syndromes using two body measurements, as well as a generalised subsystem error correction procedure.
• Published: 2011
• Publisher: University of Southern California (USC)
• Language: English
21:17 University of Southern California (USC) English 2011

#### Approximate simulation of quantum channels

We generalize the Knill-Laflamme condition to the problem of using a given noisy channel N to approximately simulate another given channel M. In particular, this yields a way to compute a near-optimal error for approximate subsystem quantum error correction. For approximate QEC, this also yield a near-optimal recovery channel. We use this result to derive a simple necessary and sufficient condition for lowest order perturbative QEC. That is, given a noise model function of a small parameter, to determine whether a given code can correct it up to lowest order in the fidelity.
• Published: 2011
• Publisher: University of Southern California (USC)
• Language: English
18:45 University of Southern California (USC) English 2011

#### Fault-tolerant quantum computation with asymmetric Bacon-Shor codes

Bacon-Shor codes are quantum subsystem codes which are constructed by combining together two quantum repetition codes, one protecting against Z (phase) errors and the other protecting against X (bit flip) errors. In many situations, for example flux qubits, the noise is biased such that Z errors are much more common than X errors; in these cases it is natural to consider an asymmetric Bacon-Shor code where the code protecting against Z errors is longer than the code protecting against X errors. This work provides fault-tolerant gadget constructions to achieve universal fault-tolerant quantum computation using asymmetric Bacon-Shor codes and controlled-Z gates as the only two-qubit gates. The qubits can be arranged in a constant-width ribbon and all gates performed on single qubits or neighboring pairs of qubits. In the presence of biased noise, these constructions allow for powerful reductions in the error rate with modest resource overhead.
• Published: 2011
• Publisher: University of Southern California (USC)
• Language: English
13:37 University of Southern California (USC) English 2011

#### Correcting noise in optical fibers via dynamic decoupling

One of the major challenges in quantum computation has been to preserve the coherence of a quantum system against dephasing effects of the environment. The information stored in photon polarization, for example, is immediately lost due to such dephasing and it is crucial to preserve the input states when one tries to transmit quantum information encoded in the photons through some communication channel. We simulate random birefringent noise along realistic lengths of optical fiber and study preservation of polarization qubits through such fibers enhanced with Carr-Purcell-Meiboom-Gill (CPMG) dynamical decoupling. The sequence, implemented with waveplates along the birefringent fiber, helps to maintain very high fidelity over a given length of the fiber. Moreover, errors arising due to the time-dependent control pulses can be completely eliminated as here one only needs to incorporate the wave plates in the prescribed way. This simple and fairly practical model is valid for preserving any general polarization state of the single photons besides providing a direction towards achieving scalable and useful quantum computation with photonic qubits.
• Published: 2011
• Publisher: University of Southern California (USC)
• Language: English
19:20 University of Southern California (USC) English 2011

#### 3D local qupit quantum code without string logical operator

Recently Haah introduced a new class of local quantum error correcting code embedded on a cubic lattice without any string logical operator. We present new codes with same property by relaxing the condition on the local particle dimension. The resulting code is well-defined when the local Hilbert space dimension is prime. These codes can be divided into two different classes: the local stabilizer generators are either symmetric or antisymmetric with respect to the inversion operation. We lower bound the number of encoded qudits by computing the commutation relation between the logical operators confined on a plane.
• Published: 2011
• Publisher: University of Southern California (USC)
• Language: English
1:02:41 University of Southern California (USC) English 2011

#### Fault-tolerant quantum computation with high threshold in two dimensions

Fault-tolerant quantum computation with high threshold in two dimensions Quantum computation is fragile. Exotic quantum states are created in the process, exhibiting entanglement among large number of particles across macroscopic distances. In realistic physical systems, decoherence acts to transform these states into more classical ones, compromising their computational power. However, decoherence can be counteracted by quantum error-correction. In my talk I first give an introduction to fault-tolerant quantum computation in the setting of two-dimensional lattices of qubits in which only nearest neighbors may interact. Such a geometric constraint is, in many physical systems considered for building a large-scale quantum computer, imposed by experimental reality. It is relevant for arrays of superconducting qubits, optical lattices and also for segmented ion traps. Efficient solutions for achieving fault-tolerance in such a scenario are topological. I will review some of the known constructions based on surface codes and color codes.
• Published: 2011
• Publisher: University of Southern California (USC)
• Language: English
58:21 University of Southern California (USC) English 2011

#### Experimental Quantum Error Correction

The Achilles' heel of quantum information processors is the fragility of quantum states and processes. Without a method to control imperfection and imprecision of quantum devices, the probability that a quantum computation succeed will decrease exponentially in the number of gates it requires. In the last fifteen years, building on the discovery of quantum error correction, accuracy threshold theorems were proved showing that error can be controlled using a reasonable amount of resources as long as the error rate is smaller than a certain threshold. We thus have a scalable theory describing how to control quantum systems. The next step is to turn this theory into practice. I will give an overview of some of the progress towards the implmentation of quantum error correction around the world with a focus on results since the last Quantum Error Correction conference at USC. I will compare the various achievements and point towards what still need to be done to get robust quantum information processors.
• Published: 2011
• Publisher: University of Southern California (USC)
• Language: English
1:00:37 University of Southern California (USC) English 2011

#### Design and analysis of autonomous quantum memories based on coherent feedback control

Key ideas from the canonical theory of quantum error correction via coding and syndrome measurement can be "pushed down" to the level of physical models with stationary Hamiltonian couplings in various ways; in this talk I will review our group's work on an approach inspired by emerging ideas in coherent-feedback quantum control, which seem well-suited to the implementation settings of nanophotonic cavity-QED and superconducting circuit-QED. I will introduce some basic ideas of continuous syndrome measurement and embedded control dynamics, and discuss a class of intuitive master equations for autonomous quantum memories that can be derived via a modeling approximation that we call the small-volume limit. I will discuss some computational challenges involved in constructing and integrating such master equations for memories based on complex codes, and close with some general remarks on coherent feedback as a tool for interaction synthesis and disturbance attenuation in quantum engineering.
• Published: 2011
• Publisher: University of Southern California (USC)
• Language: English
1:08:37 University of Southern California (USC) English 2011

#### Dynamical Decoupling a tutorial

Dynamical decoupling (DD) is an open-loop method for decoherence reduction based on the frequent application of strong pulses, designed to cancel terms in the system-bath interaction Hamiltonian. DD has been tested and successfully implemented in numerous experimental systems. After introducing the basics of DD, this tutorial will review recent developments concerning optimized pulse sequences, capable of canceling the system-bath interaction to arbitrary given order. The integration of DD with computation will also be addressed, from the complementary perspectives of "decouple while compute" and "decouple then compute". The former requires a commuting set of pulses and gates, which can be designed using tools borrowed from quantum error correction. The latter allows DD to be integrated with fault tolerant quantum computation.
• Published: 2011
• Publisher: University of Southern California (USC)
• Language: English
44:45 University of Southern California (USC) English 2011

#### Fault-tolerant quantum computation via adiabatic holonomies

I will describe various methods for realizing fault-tolerant quantum computation in terms of adiabatic geometric transformations.
• Published: 2011
• Publisher: University of Southern California (USC)
• Language: English
16:51 University of Southern California (USC) English 2011

#### Entanglement-assisted quantum LDPC codes from combinatorial designs

The entanglement-assisted stabilizer formalism is a generalized form of the stabilizer formalism. This framework allows the code designer to take advantage of a significantly wider range of classical error-correcting codes by using pairs of qubits in a maximally entangled state (or ebits). Low-density parity-check (LDPC) codes are among the best known error-correcting codes in terms of error correction performance and decoding complexity in the classical domain and can also be imported to the quantum domain in a simple manner through the entanglement-assisted stabilizer formalism. From a practical viewpoint, it is desirable to rely on fewer ebits while keeping the error correction ability inherited from classical LDPC codes. We present necessary and sufficient conditions for the existence of quantum LDPC codes consuming only one ebit which are obtainable from pairs of identical LDPC codes, and show relations of entanglement-assisted quantum LDPC codes to some fundamental classes of combinatorial designs.
• Published: 2011
• Publisher: University of Southern California (USC)
• Language: English
25:22 University of Southern California (USC) English 2011

#### Approximate Operator Quantum Error Correction

Operator quantum error correction (OQEC) extends the standard formalism of quantum error correction (QEC) to codes in which only a subsystem within a subspace of states is used to store information in a noise-resilient fashion. Motivated by recent work on approximate QEC, which makes it possible to construct subspace codes beyond the framework of perfect error correction, we investigate the problem of approximate operator quantum error correction (AOQEC). We demonstrate easily checkable sufficient conditions for the existence of approximate subsystem codes. Furthermore, we prove the efficacy of the transpose channel as a simple-to-construct recovery map that works nearly as well as the optimal recovery channel, with optimality defined in terms of worst-case fidelity over all code states. This work generalizes our earlier approach of using the transpose channel for approximate subspace correction to the case of approximate OQEC, thus bringing us closer to a full analytical understanding of approximate codes.
• Published: 2011
• Publisher: University of Southern California (USC)
• Language: English
18:07 University of Southern California (USC) English 2011

#### Bounds on achievable rates of sparse quantum codes used over the quantum erasure channel

We study the performance of locally decodable sparse quantum codes. The most familiar example of such a code is Kitaev’s toric code. During the last ten years, a number of different constructions of these codes appeared, for example: surfaces codes, finite geometry codes or Latin square codes. These codes are defined by stabilizer group with generators of low weight. If the stabilizer matrix has row weight m and column weight l, we talk about a (l,m) code. Our main result is an upper bound on achievable rates of stabilizer (l,m) codes, as a function of m and l. Achievable rates are rates for which decoding is possible with high probability.
• Published: 2011
• Publisher: University of Southern California (USC)
• Language: English
38:57 University of Southern California (USC) English 2011

#### Decoding algorithms for topological codes

I will talk about the problem of decoding a topological code, that consists of identifying the optimal recovery operation given the syndrome of an error, or equivalently of inferring the most likely world-line homology given a defect configuration. I will describe a new decoding algorithm [Phys. Rev. Lett. 104 050504 arXiv:0911.0581 and arXiv:1006.1362] for Kitaev's toric code (KTC) that runs in a time proportional to the log of the number of particles, an improvement over the previously known polynomial-time decoding algorithm. This algorithm also achieves a higher threshold on the depolarizing channel. Moreover, we have recently shown that all two dimensional topological stabilizer codes can be mapped onto each other by local transformations [arXiv:1103.4606, arXiv:1107.2707]. This local mapping enables us to use any decoding algorithm suitable for one of these codes to decode other codes in the same topological phase. We illustrate this idea with the topological color code that is found to be locally equivalent to two copies of KTC and we extend it to decode the topological subsystem color code.
• Published: 2011
• Publisher: University of Southern California (USC)
• Language: English
39:44 University of Southern California (USC) English 2011

#### Detecting Incapacity

Using unreliable or noisy components for reliable communication requires error correction. But which noise processes can support information transmission, and which are too destructive? For classical systems any channel whose output depends on its input has the capacity for communication, but the situation is substantially more complicated in the quantum setting. We find a generic test for incapacity based on any suitable forbidden transformation---a protocol for communication with a channel passing our test would also allow us to implement the associated forbidden transformation. Our approach includes both known quantum incapacity tests -positive partial transposition (PPT) and antidegradability (no cloning)- as special cases, putting them both on the same footing. We also find a physical principle explaining the nondistillability of PPT states: Any protocol for distilling entanglement from such a state would also give a protocol for implementing the forbidden time-reversal operation.
• Published: 2011
• Publisher: University of Southern California (USC)
• Language: English
32:11 University of Southern California (USC) English 2011

#### Local stabilizer codes in three dimensions without string logical operators

We report examples of local stabilizer codes in 3D that have no string logical operators. Previously known local stabilizer codes in 3D all have string-like logical operators, which make the codes non-self-correcting. We introduce a notion of "logical string segments" to avoid difficulties in defining one dimensional objects in discrete lattices. We prove that every string-like logical operator of our code can be deformed to a disjoint union of short segments, and each segment is in the stabilizer group. The code space dimension depends on the number-theoretic property of the system size.
• Published: 2011
• Publisher: University of Southern California (USC)
• Language: English
1:03:48 University of Southern California (USC) English 2011

#### Fault-tolerant quantum computing

This is a tutorial on fault-tolerant quantum computing, with an emphasis on concatenated-coding and topological-coding approaches in the quantum circuit model. The tutorial will cover key protocols used in fault-tolerant quantum computing schemes, including syndrome extraction, syndrome decoding, and encoded computation with magic states. It will also cover well-established analysis techniques, including extended rectangles and Monte Carlo simulation. A basic familiarity with stabilizer-based quantum error correction is assumed.
• Published: 2011
• Publisher: University of Southern California (USC)
• Language: English
43:55 University of Southern California (USC) English 2011

#### Long coherence times with dense trapped atomics

Atomic ensembles have many potential applications in quantum information science. Owing to collective enhancement, working with ensembles at high densities increases the overall efficiency of quantum operations, but at the same time also increases the collision rate and markedly changes the time dynamics of a stored coherence. We study theoretically and experimentally the coherent dynamics of cold atoms under these conditions. A closed form expression for the spectral line shape is derived for discrete fluctuations in terms of the bare spectrum and the Poisson rate constant of collisions, which deviates from the canonical stochastic theory of Kubo. We measure a prolongation of the coherence times of optically trapped rubidium atoms as their density increases, a phenomenon we call collisional narrowing in analog to the well known motional narrowing effect in NMR. We explain under what circumstances collisional narrowing can be transformed into collisional broadening. On account of collisions, conventional echo techniques fail to suppress this dephasing, and multi-pulse dynamical decoupling sequences are required. We present experiments demonstrating a 20-fold increase of the coherence time when a sequence with more than 200 pi pulses is applied. We perform quantum process tomography and demonstrate that using the decoupling scheme a dense ensemble with an optical depth of >200 can be used as an atomic memory with coherence times exceeding 3 sec. Further optimization requires utilizing specific features of the collisional bath, which we measure directly.
• Published: 2011
• Publisher: University of Southern California (USC)
• Language: English
43:28 University of Southern California (USC) English 2011

#### Smallest Errors achievable by Dynamical Decoupling (and How to Maintain Them)

We explore the fundamental limits on coherence preservation by dynamical decoupling methods in terms of control time scales and the spectrum/bandwidth of the environment. We focus on a decohering qubit controlled by arbitrary sequences of pi pulses. Using results from mathematical analysis, we establish a lower bound for coherence loss in terms of the minimum time between the pulses and the spectral cutoff frequency of the environment. We argue that similar bounds are applicable to a variety of open-loop unitary control methods while we find no explicit dependence of such lower bounds on the total control time. We use these findings to automatically generate "bandwidth adapted dynamical decoupling" sequences that can be used for preserving a qubit up to arbitrary times with the best fidelities theoretically possible given the available control capabilities. We also introduce "Walsh dynamical decoupling" schemes that are optimized for digital sequence generation. Our results imply that fact that, unlike in quantum fault-tolerant architecture, errors cannot be reduced indefinitely using reversible control methods yet a small error can be maintained for a long time.
• Published: 2011
• Publisher: University of Southern California (USC)
• Language: English
41:25 University of Southern California (USC) English 2011

#### Towards practical classical processing for the surface code

The surface code is unarguably the leading QEC code, featuring a high threshold error rate ~1%, low overhead implementations of the entire Clifford group, and flexible, arbitrarily long-range logical gates - all despite only requiring a 2-D lattice of qubits with NN interactions. These highly desirable features come at the cost of high classical processing complexity. We show how to perform the processing associated with an n by n lattice of qubits, each being manipulated fault-tolerantly, in O(n^2) average time per QEC round. We describe how to parallelize the algorithm to O(1), using constant computing resources per unit area and local communication. Both complexities are optimal.
• Published: 2011
• Publisher: University of Southern California (USC)
• Language: English
39:06 University of Southern California (USC) English 2011

#### Universal topological phase of 2D stabilizer codes

We show that 2D topological stabilizer codes can all be characterized in terms of a finite number of charges and string operators. This is true either for subspace or subsystem codes, and it has direct applications for error correction. Subspace codes are directly connected to topologically ordered systems, and we show that all 2D topological stabilizer codes are locally equivalent to several copies of one universal phase: Kitaev's topological code.
• Published: 2011
• Publisher: University of Southern California (USC)
• Language: English
1:30:04 University of Southern California (USC) English 2011

#### Panel moderated by Byrd

• Published: 2011
• Publisher: University of Southern California (USC)
• Language: English
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