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

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

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

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