1,726 research outputs found

### Bounds for Approximation in Total Variation Distance by Quantum Circuits

It was recently shown that for reasonable notions of approximation of states
and functions by quantum circuits, almost all states and functions are
exponentially hard to approximate [Knill 1995]. The bounds obtained are
asymptotically tight except for the one based on total variation distance
(TVD). TVD is the most relevant metric for the performance of a quantum
circuit. In this paper we obtain asymptotically tight bounds for TVD. We show
that in a natural sense, almost all states are hard to approximate to within a
TVD of 2/e-\epsilon even for exponentially small \epsilon. The quantity 2/e is
asymptotically the average distance to the uniform distribution. Almost all
states with probability amplitudes concentrated in a small fraction of the
space are hard to approximate to within a TVD of 2-\epsilon. These results
imply that non-uniform quantum circuit complexity is non-trivial in any
reasonable model. They also reinforce the notion that the relative information
distance between states (which is based on the difficulty of transforming one
state to another) fully reflects the dimensionality of the space of qubits, not
the number of qubits.Comment: uuencoded compressed postscript, LACES 68Q-95-3

### Non-binary Unitary Error Bases and Quantum Codes

Error operator bases for systems of any dimension are defined and natural
generalizations of the bit/sign flip error basis for qubits are given. These
bases allow generalizing the construction of quantum codes based on eigenspaces
of Abelian groups. As a consequence, quantum codes can be constructed from
linear codes over \ints_n for any $n$. The generalization of the punctured
code construction leads to many codes which permit transversal (i.e. fault
tolerant) implementations of certain operations compatible with the error
basis.Comment: 10 pages, preliminary repor

### Approximation by Quantum Circuits

In a recent preprint by Deutsch et al. [1995] the authors suggest the
possibility of polynomial approximability of arbitrary unitary operations on
$n$ qubits by 2-qubit unitary operations. We address that comment by proving
strong lower bounds on the approximation capabilities of g-qubit unitary
operations for fixed g. We consider approximation of unitary operations on
subspaces as well as approximation of states and of density matrices by quantum
circuits in several natural metrics. The ability of quantum circuits to
probabilistically solve decision problem and guess checkable functions is
discussed. We also address exact unitary representation by reducing the upper
bound by a factor of n^2 and by formalizing the argument given by Barenco et
al. [1995] for the lower bound. The overall conclusion is that almost all
problems are hard to solve with quantum circuits.Comment: uuencoded, compressed postscript, LACES 68Q-95-2

### Group Representations, Error Bases and Quantum Codes

This report continues the discussion of unitary error bases and quantum codes
begun in "Non-binary Unitary Error Bases and Quantum Codes". Nice error bases
are characterized in terms of the existence of certain characters in a group. A
general construction for error bases which are non-abelian over the center is
given. The method for obtaining codes due to Calderbank et al. is generalized
and expressed purely in representation theoretic terms. The significance of the
inertia subgroup both for constructing codes and obtaining the set of
transversally implementable operations is demonstrated.Comment: 11 pages, preliminary repor

### How to upload a physical state to the correlation space

In the framework of the computational tensor network [D. Gross and J. Eisert,
Phys. Rev. Lett. {\bf98}, 220503 (2007)], the quantum computation is performed
in a virtual linear space which is called the correlation space. It was
recently shown [J. M. Cai, W, D\"ur, M. Van den Nest, A. Miyake, and H. J.
Briegel, Phys. Rev. Lett. {\bf103}, 050503 (2009)] that a state in the
correlation space can be downloaded to the real physical space. In this letter,
conversely, we study how to upload a state from a real physical space to the
correlation space being motivated by the virtual-real hybrid quantum
information processing. After showing the impossibility of the cloning of a
state between the real physical space and the correlation space, we propose a
simple teleportation-like method of the upload. Applications of this method
also enable the Gottesman-Chuang gate teleportation trick and the entanglement
swapping in the virtual-real hybrid setting. Furthermore, compared with the
inverse of the downloading method by Cai, et. al., which also works as the
upload, our uploading method has several advantages.Comment: 8 pages, 4 figure

### A Note on Linear Optics Gates by Post-Selection

Recently it was realized that linear optics and photo-detectors with feedback
can be used for theoretically efficient quantum information processing. The
first of three steps toward efficient linear optics quantum computation (eLOQC)
was to design a simple non-deterministic gate, which upon post-selection based
on a measurement result implements a non-linear phase shift on one mode. Here a
computational strategy is given for finding non-deterministic gates for bosonic
qubits with helper photons. A more efficient conditional sign flip gate is
obtained.Comment: 14 pages. Minor changes for clarit

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