Set Of Unitary Matrices Research Articles

RETRACTED ARTICLE: Noise tailoring for quantum circuits via unitary 2t-design

Because of environmental variations and imperfect operations, real-world quantum computers produce different coherent errors that are difficult to estimate. Here, we propose a method whereby the twirled noise over a unitary 2t-design (a set of unitary matrices that approximate the entire unitary group) for quantum circuits can be tailored into stochastic noise. Then, we prove that local random circuits for twirling separable noisy channel over the Clifford group can be used to construct a unitary 2t-design, which is easy to implement in experiments. Moreover, we prove that our method is robust to gate-dependent and gate-independent noise. The stochastic noise can be both estimated by average fidelity and directly obtained by randomized benchmarking via unitary 2t-designs. Obtaining such tailored noise is an important guarantee for achieving fault-tolerant quantum computation.

A New Quaternary Alphabet Signal Constellation for Differential Unitary Space-Time Modulation

In this letter, a new constellation design for differential unitary space-time modulation is proposed. We generate a set of unitary matrices from the multiplication of a simple unitary matrix with entries of ± 1 by diagonal matrices made from quaternary alphabets. Then, the constellation is formed from the product of the optimum matrix chosen from this set and the optimized diagonal matrix first used for the cyclic codes. We show that, without deteriorating the transmission rate, the proposed constellation yields a larger diversity product as well as a lower union bound for pairwise error probability with the same or a lower complexity compared to other previously proposed constellations.

The sum of orthogonal matrices

Let F∈{R,C,H}. Let Un(F) be the set of unitary matrices in Mn(F), and let On(F) be the set of orthogonal matrices in Mn(F). Suppose n⩾2. We show that every A∈Mn(F) can be written as a sum of matrices in Un(F) and of matrices in On(F). Let A∈Mn(F) be given and let k⩾2 be the least integer that is a least upper bound of the singular values of A. When F=C, we show that A can be written as a sum of k matrices from Un(F). When F=R, we show that if k⩽3, then A can be written as a sum of 6 orthogonal matrices; if k⩾4, we show that A can be written as a sum of k+2 orthogonal matrices.

An Application of the Bruhat Decomposition to the Design of Full Diversity Unitary Space–Time Codes

A full diversity constellation, that is, a set of unitary matrices whose differences have nonzero determinant, is a design criterion for codes with good performance using differential unitary space-time modulation. Fixed-point free groups and the infinite group <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SU</i> (2) have been used to produce full diversity unitary group constellations. In this paper, we present a new Bruhat decomposition design for constructing full diversity unitary space-time constellations for any number of antennas. They are constructed from cosets of a unitary diagonal subgroup <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</i> , and our design has a particularly simple structure for the case where the number of transmitter antennas is prime. We also consider the extension of these constellation designs for an even number of transmitter antennas by replacing <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</i> with a Hamiltonian constellation. Some examples of proposed constellations for two to six transmitter antennas are given. Simulations show that our proposed constellations perform well in unknown Rayleigh fading channel.

Evenly distributed unitaries: On the structure of unitary designs

We clarify the mathematical structure underlying unitary t-designs. These are sets of unitary matrices, evenly distributed in the sense that the average of any tth order polynomial over the design equals the average over the entire unitary group. We present a simple necessary and sufficient criterion for deciding if a set of matrices constitutes a design. Lower bounds for the number of elements of 2-designs are derived. We show how to turn mutually unbiased bases into approximate 2-designs whose cardinality is optimal in leading order. Designs of higher order are discussed and an example of a unitary 5-design is presented. We comment on the relation between unitary and spherical designs and outline methods for finding designs numerically or by searching character tables of finite groups. Further, we sketch connections to problems in linear optics and questions regarding typical entanglement.

Interpolation Based Unitary Precoding for Spatial Multiplexing MIMO-OFDM With Limited Feedback

Spatial multiplexing with linear precoding is a simple technique for achieving high spectral efficiency in multiple-input multiple-output orthogonal frequency division multiplexing (MIMO-OFDM) systems. Linear precoding requires channel state information for each OFDM subcarrier, which can be achieved using feedback. To reduce the amount of feedback, this paper proposes a limited feedback architecture that combines precoder quantization with a special matrix interpolator. In the proposed system, the receiver sends information about a fraction of the precoding matrices to the transmitter and the transmitter reconstructs the precoding matrices for all the subcarriers. A new interpolator is proposed inspired by spherical interpolation that respects the orthogonal columns of the precoding matrices and the performance invariance to right multiplication by a unitary matrix. The interpolator is parameterized by a set of unitary matrices; a construction of a suitable set is briefly described. Simulations illustrate the performance of limited feedback precoding with coding, estimation or prediction error, and time variation for bit error rate (BER), mutual information, and mean squared error (MSE)

Differential space-frequency trellis codes

A new differential space-frequency trellis coding (DSFTC) scheme is proposed for multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) systems. The transmitter differentially encodes the message drawn from a set of unitary matrices following a trellis encoding rule. At the receiver, where no channel state information (CSI) is available, differential decoding and soft-input Viterbi algorithms are employed in the decoding process. The proposed codes can achieve full rate, 1 symbol/subcarrier, and full spatial diversity. Simulation results show that the proposed codes provide significant improvements over differential space-frequency block codes (DSFBC) due to increased coding gains

QPSK differential space time coding on different unitary matrices sets and initializations

AbstractThis paper investigates a distinct set of complex unitary matrices for differential space time coding by using QPSK modulation. The numerical results show that the proper selection of the initial transmission matrix and the set of unitary matrices can efficiently improve the bit error rate performance, especially for the antennas correlated fading channel. The computer simulations are evaluated over slow and fast Rayleigh fading channels. Copyright © 2005 John Wiley &amp; Sons, Ltd.

QPSK Differential Space Time Coding on Different Unitary Matrices Sets and Initializations

This letter investigates a distinct set of complex unitary matrices for differential space time coding by using QPSK modulation. The numerical results show that the properly selection of the initial transmission matrix and the set of unitary matrices can efficiently improve the bit error rate (BER) performance, especially for the antennas correlated fading channel. The computer simulations are evaluated over slow and fast Rayleigh fading channels.

Three-transmit-antenna space-time codes based on SU(3)

Fully diverse constellations, i.e., a set of unitary matrices whose pairwise differences are nonsingular, are useful in multiantenna communications especially in multiantenna differential modulation, since they have good pairwise error properties. Recently, group theoretic ideas, especially fixed-point-free (fpf) groups, have been used to design fully diverse constellations of unitary matrices. Here, we give systematic design methods of space-time codes which are appropriate for three-transmit-antenna differential modulation. The structures of the codes are motivated by the special unitary Lie group SU(3). One of the codes, which is called the AB code, has a fast maximum-likelihood (ML) decoding algorithm using complex sphere decoding. Diversity products of the codes can be easily calculated, and simulated performance shows that they are better than group-based codes, especially at high rates and as good as the elaborately designed nongroup code.

Design of Fully Diverse Multiple-Antenna Codes Based on&amp;lt;tex&amp;gt;$Sp(2)$&amp;lt;/tex&amp;gt;

Fully diverse constellations, i.e., sets of unitary matrices whose pairwise differences are nonsingular, are useful in multiple-antenna communications, especially in multiple-antenna differential modulation, since they have good pairwise error properties. Recently, group theoretic ideas, especially fixed-point-free (fpf) groups, have been used to design fully diverse constellations of unitary matrices. Here we construct four-transmit-antenna constellations appropriate for differential modulation based on the symplectic group Sp(2). They can be regarded as extensions of Alamouti's celebrated two-transmit-antenna orthogonal design which can be constructed from the group Sp(1). We further show that the structure of Sp(2) codes lends itself to efficient maximum-likelihood (ML) decoding via the sphere decoding algorithm. Finally, the performance of Sp(2) codes is compared with that of other existing codes including Alamouti's orthogonal design, a 4/spl times/4 complex orthogonal design, Cayley differential unitary space-time codes and group-based codes.

Representation theory for high-rate multiple-antenna code design

Multiple antennas can greatly increase the data rate and reliability of a wireless communication link in a fading environment, but the practical success of using multiple antennas depends crucially on our ability to design high-rate space-time constellations with low encoding and decoding complexity. It has been shown that full transmitter diversity, where the constellation is a set of unitary matrices whose differences have nonzero determinant, is a desirable property for good performance. We use the powerful theory of fixed-point-free groups and their representations to design high-rate constellations with full diversity. Furthermore, we thereby classify all full-diversity constellations that form a group, for all rates and numbers of transmitter antennas. The group structure makes the constellations especially suitable for differential modulation and low-complexity decoding algorithms. The classification also reveals that the number of different group structures with full diversity is very limited when the number of transmitter antennas is large and odd. We, therefore, also consider extensions of the constellation designs to nongroups. We conclude by showing that many of our designed constellations perform excellently on both simulated and real wireless channels.