134 research outputs found
Multiple Particle Interference and Quantum Error Correction
The concept of multiple particle interference is discussed, using insights
provided by the classical theory of error correcting codes. This leads to a
discussion of error correction in a quantum communication channel or a quantum
computer. Methods of error correction in the quantum regime are presented, and
their limitations assessed. A quantum channel can recover from arbitrary
decoherence of x qubits if K bits of quantum information are encoded using n
quantum bits, where K/n can be greater than 1-2 H(2x/n), but must be less than
1 - 2 H(x/n). This implies exponential reduction of decoherence with only a
polynomial increase in the computing resources required. Therefore quantum
computation can be made free of errors in the presence of physically realistic
levels of decoherence. The methods also allow isolation of quantum
communication from noise and evesdropping (quantum privacy amplification).Comment: Submitted to Proc. Roy. Soc. Lond. A. in November 1995, accepted May
1996. 39 pages, 6 figures. This is now the final version. The changes are
some added references, changed final figure, and a more precise use of the
word `decoherence'. I would like to propose the word `defection' for a
general unknown error of a single qubit (rotation and/or entanglement). It is
useful because it captures the nature of the error process, and has a verb
form `to defect'. Random unitary changes (rotations) of a qubit are caused by
defects in the quantum computer; to entangle randomly with the environment is
to form a treacherous alliance with an enemy of successful quantu
Cryptography from tensor problems
We describe a new proposal for a trap-door one-way function. The new proposal belongs to the "multivariate quadratic" family but the trap-door is different from existing methods, and is simpler
Towards Robust Quantum Computation
Quantum computation is a subject of much theoretical promise, but has not
been realized in large scale, despite the discovery of fault-tolerant
procedures to overcome decoherence. Part of the reason is that the
theoretically modest requirements still present daunting experimental
challenges. The goal of this Dissertation is to reduce various resources
required for robust quantum computation, focusing on quantum error correcting
codes and solution NMR quantum computation. A variety of techniques have been
developed, including high rate quantum codes for amplitude damping, relaxed
criteria for quantum error correction, systematic construction of
fault-tolerant gates, recipes for quantum process tomography, techniques in
bulk thermal state computation, and efficient decoupling techniques to
implement selective coupled logic gates. A detailed experimental study of a
quantum error correcting code in NMR is also presented. The Dissertation
clarifies and extends results previously reported in quant-ph/9610043,
quant-ph/9704002, quant-ph/9811068, quant-ph/9904100, quant-ph/9906112,
quant-ph/0002039. Additionally, a procedure for quantum process tomography
using maximally entangled states, and a review on NMR quantum computation are
included.Comment: 243 pages, PhD Dissertation, Stanford University, July 200
Quantum expanders and the quantum entropy difference problem
We define quantum expanders in a natural way. We show that under certain
conditions classical expander constructions generalize to the quantum setting,
and in particular so does the Lubotzky, Philips and Sarnak construction of
Ramanujan expanders from Cayley graphs of the group PGL. We show that this
definition is exactly what is needed for characterizing the complexity of
estimating quantum entropies.Comment: 30 pages, 1 figur
Tailoring surface codes: Improvements in quantum error correction with biased noise
For quantum computers to reach their full potential will require error correction. We study the surface code, one of the most promising quantum error correcting codes, in the context of predominantly dephasing (Z-biased) noise, as found in many quantum architectures. We find that the surface code is highly resilient to Y-biased noise, and tailor it to Z-biased noise, whilst retaining its practical features. We demonstrate ultrahigh thresholds for the tailored surface code: ~39% with a realistic bias of = 100, and ~50% with pure Z noise, far exceeding known thresholds for the standard surface code: ~11% with pure Z noise, and ~19% with depolarizing noise. Furthermore, we provide strong evidence that the threshold of the tailored surface code tracks the hashing bound for all biases. We reveal the hidden structure of the tailored surface code with pure Z noise that is responsible for these ultrahigh thresholds. As a consequence, we prove that its threshold with pure Z noise is 50%, and we show that its distance to Z errors, and the number of failure modes, can be tuned by modifying its boundary. For codes with appropriately modified boundaries, the distance to Z errors is O(n) compared to O(n1/2) for square codes, where n is the number of physical qubits. We demonstrate that these characteristics yield a significant improvement in logical error rate with pure Z and Z-biased noise. Finally, we introduce an efficient approach to decoding that exploits code symmetries with respect to a given noise model, and extends readily to the fault-tolerant context, where measurements are unreliable. We use this approach to define a decoder for the tailored surface code with Z-biased noise. Although the decoder is suboptimal, we observe exceptionally high fault-tolerant thresholds of ~5% with bias = 100 and exceeding 6% with pure Z noise. Our results open up many avenues of research and, recent developments in bias-preserving gates, highlight their direct relevance to experiment
A survey on the complexity of learning quantum states
We survey various recent results that rigorously study the complexity of
learning quantum states. These include progress on quantum tomography, learning
physical quantum states, alternate learning models to tomography and learning
classical functions encoded as quantum states. We highlight how these results
are paving the way for a highly successful theory with a range of exciting open
questions. To this end, we distill 25 open questions from these results.Comment: Invited article by Nature Review Physics. 39 pages, 6 figure
The Study of Entangled States in Quantum Computation and Quantum Information Science
This thesis explores the use of entangled states in quantum computation and
quantum information science. Entanglement, a quantum phenomenon with no
classical counterpart, has been identified as an important and quantifiable
resource in many areas of theoretical quantum information science, including
quantum error correction, quantum cryptography, and quantum algorithms. We
first investigate the equivalence classes of a particular class of entangled
states (known as graph states due to their association with mathematical
graphs) under local operations. We prove that for graph states corresponding to
graphs with neither cycles of length 3 nor 4, the equivalence classes can be
characterized in a very simple way. We also present software for analyzing and
manipulating graph states. We then study quantum error-correcting codes whose
codewords are highly entangled states. An important area of investigation
concerning QECCs is to determine which resources are necessary in order to
carry out any computation on the code to an arbitrary degree of accuracy, while
simultaneously maintaining a high degree of resistance to noise. We prove that
transversal gates, which are designed to prevent the propagation of errors
through a system, are insufficient to achieve universal computation on almost
all QECCs. Finally, we study the problem of creating efficient quantum circuits
for creating entangling measurements. Entangling measurements can be used to
harness the apparent extra computing power of quantum systems by allowing us to
extract information about the global, collective properties of a quantum state
using local measurements. We construct explicit quantum circuits that create
entangling measurements, and show that these circuits scale polynomially in the
input parameters.Comment: 206 pages, 42 figure
Robust encoding of a qubit in a molecule
We construct quantum error-correcting codes that embed a finite-dimensional
code space in the infinite-dimensional Hilbert state space of rotational states
of a rigid body. These codes, which protect against both drift in the body's
orientation and small changes in its angular momentum, may be well suited for
robust storage and coherent processing of quantum information using rotational
states of a polyatomic molecule. Extensions of such codes to rigid bodies with
a symmetry axis are compatible with rotational states of diatomic molecules, as
well as nuclear states of molecules and atoms. We also describe codes
associated with general nonabelian compact Lie groups and develop orthogonality
relations for coset spaces, laying the groundwork for quantum information
processing with exotic configuration spaces.Comment: 28(+15) pages, 6 figures, 5 tables; v2 minor change
Quantum algorithms, symmetry, and Fourier analysis
I describe the role of symmetry in two quantum algorithms, with a focus on how that symmetry is made manifest by the Fourier transform. The Fourier transform can be considered in a wider context than the familiar one of functions on Rn or Z/nZ; instead it can be defined for an arbitrary group where it is known as representation theory. The first quantum algorithm solves an instance of the hidden subgroup problem--distinguishing conjugates of the Borel subgroup from each other in groups related to PSL(2; q). I use the symmetry of the subgroups under consideration to reduce the problem to a mild extension of a previously solved problem. This generalizes a result of Moore, Rockmore, Russel and Schulman[33] by switching to a more natural measurement that also applies to prime powers. In contrast to the first algorithm, the second quantum algorithm is an attempt to use naturally continuous spaces. Quantum walks have proved to be a useful tool for designing quantum algorithms. The natural equivalent to continuous time quantum walks is evolution with the Schrodinger equation, under the kinetic energy Hamiltonian for a massive particle. I take advantage of quantum interference to find the center of spherical shells in high dimensions. Any implementation would be likely to take place on a discrete grid, using the ability of a digital quantum computer to simulate the evolution of a quantum system. In addition, I use ideas from the second algorithm on a different set of starting states, and find that quantum evolution can be used to sample from the evolute of a plane curve. The method of stationary phase is used to determine scaling exponents characterizing the precision and probability of success for this procedure
- …