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arXiv:1105.1464 · International Journal of Quantum Information

QUANTUM HIERARCHIC MODELS FOR INFORMATION PROCESSING

M. V. Altaĭsky, N. E. Kaputkina

Published Mar 1, 2012• 1 citation
arXivHTMLPDF

Abstract

Both classical and quantum computations operate with the registers of bits. At nanometer scale the quantum fluctuations at the position of a given bit, say, a quantum dot, not only lead to the decoherence of quantum state of this bit, but also affect the quantum states of the neighboring bits, and therefore affect the state of the whole register. That is why the requirement of reliable separate access to each bit poses the limit on miniaturization, i.e. constrains the memory capacity and the speed of computation. In the present paper we suggest an algorithmic way to tackle the problem of constructing reliable and compact registers of quantum bits. We suggest accessing the states of a quantum register hierarchically, descending from the state of the whole register to the states of its parts. Our method is similar to quantum wavelet transform, and can be applied to information compression, quantum memory, quantum computations.

DOI
10.1142/s0219749912500268

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Both classical and quantum computations operate with the registers of bits. At nanometer scale the quantum fluctuations at the position of a given bit, say, a quantum dot, not only lead to the decoherence of quantum state of this bit, but also affect the quantum states of the neighboring bits, and therefore affect the state of the whole register. That is why the requirement of reliable separate access to each bit poses the limit on miniaturization, i.e. constrains the memory capacity and the speed of computation. In the present paper we suggest an algorithmic way to tackle the problem of constructing reliable and compact registers of quantum bits. We suggest accessing the states of a quantum register hierarchically, descending from the state of the whole register to the states of its parts. Our method is similar to quantum wavelet transform, and can be applied to information compression, quantum memory, quantum computations.
The quantum Fourier transform (QFT), a quantum analog of the classical Fourier transform, has been shown to be a powerful tool in developing quantum algorithms. However, in classical computing there is another class of unitary transforms, the wavelet transforms, which are every bit as useful as the Fourier transform. Wavelet transforms are used to expose the multi-scale structure of a signal and are likely to be useful for quantum image processing and quantum data compression. In this paper, we derive efficient, complete, quantum circuits for two representative quantum wavelet transforms, the quantum Haar and quantum Daubechies D(4)D^{(4)}D(4) transforms. Our approach is to factor the operators for these transforms into direct sums, direct products and dot products of unitary matrices. In so doing, we find that permutation matrices, a particular class of unitary matrices, play a pivotal role. Surprisingly, we find that operations that are easy and inexpensive to implement classically are not always easy and inexpensive to implement quantum mechanically, and vice versa. In particular, the computational cost of performing certain permutation matrices is ignored classically because they can be avoided explicitly. However, quantum mechanically, these permutation operations must be performed explicitly and hence their cost enters into the full complexity measure of the quantum transform. We consider the particular set of permutation matrices arising in quantum wavelet transforms and develop efficient quantum circuits that implement them. This allows us to design efficient, complete quantum circuits for the quantum wavelet transform.
Abstract Wavelet transforms are widely used in various fields of science and engineering as a mathematical tool with features that reveal information ignored by the Fourier transform. Unlike the Fourier transform, which is unique, a wavelet transform is specified by a sequence of numbers associated with the type of wavelet used and an order parameter specifying the length of the sequence. While the quantum Fourier transform, a quantum analog of the classical Fourier transform, has been pivotal in quantum computing, prior works on quantum wavelet transforms (QWTs) were limited to the second and fourth order of a particular wavelet, the Daubechies wavelet. Here we develop a simple yet efficient quantum algorithm for executing any wavelet transform on a quantum computer. Our approach is to decompose the kernel matrix of a wavelet transform as a linear combination of unitaries (LCU) that are compilable by easy-to-implement modular quantum arithmetic operations and use the LCU technique to construct a probabilistic procedure to implement a QWT with a known success probability. We then use properties of wavelets to make this approach deterministic by a few executions of the amplitude amplification strategy. We extend our approach to a multilevel wavelet transform and a generalized version, the packet wavelet transform, establishing computational complexities in terms of three parameters: the wavelet order M , the dimension N of the transformation matrix, and the transformation level d . We show the cost is logarithmic in N , linear in d and superlinear in M . Moreover, we show the cost is independent of M for practical applications. Our proposed QWTs could be used in quantum computing algorithms in a similar manner to their well-established counterpart, the quantum Fourier transform.
Algorithms such as quantum wavelet transform (QWT) can be utilized on quantum processors to gain significant speedup compared to their classical counterparts. Domains such as high-energy-physics and remote-sensing hyperspectral imagery require high compute capabilities and can benefit from applying QWT for dimension reduction of multidimensional data. However, quantum circuits for QWT include permutations that contribute to large overall circuit depth. Deep circuits pose a critical challenge for state-of-the-art quantum computers because of quantum decoherence. In this work, we propose QWT circuits, which are optimized in terms of circuit depth, to account for the effects of decoherence, and resulting in high fidelity and efficient implementation on quantum processors. We present the circuits in generalized forms and show that they can be used for multilevel decomposable, multidimensional wavelet operations. Experimental evaluation of the proposed circuits is performed through simulation using MATLAB and IBM-Q qasm, and implemented on a real 15-qubit quantum processor from IBM.
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Abstract In this paper, a shift register (SR) memory based on rapid single flux quantum (RSFQ) logic for bit-parallel access is proposed. The memory cell changes from a row in the memory array to a column to increase the bit of data accessed per one clock cycle. The design of the 8 bit × 2 bit (8 × 2) memory is introduced. The memory consists of four parts: a SR memory array, an input buffer, an output buffer, and a controller, in which the controller is used to produce a control signal at a certain clock cycle specified by the address data. We have obtained the low speed test of the 8 × 2 memory fabricated by 6 kA cm −2 Nb process (SIMIT-Nb03), all bit-parallel access operations (write, read, clear and write, and read again) have been verified. We also simulated the 8 × 8 memory with data width expansion and obtained the maximum operating frequency of 34 GHz after the place and route with the SIMIT-Nb03 cell library. Based on the hierarchical storage, we have used the 8 × 8 memory as a basic block to construct large size of memory system, the access time and bandwidth have been evaluated. The bit-parallel memory shows higher bandwidth than bit-serial memory with the same capacity, especially for large capacity memory.
We are interested in fundamental limits to computation imposed by physical constraints. In particular, the physical laws of motion constrain the speed at which a computer can transition between well-defined states. Here, we discuss speed limits in the context of quantum computing. We review some relevant parts of the theory of Finsler metrics on Lie groups and homogeneous spaces such as the special unitary groups and complex projective spaces. We show how these constructions can be applied to analysing the limit to the speed of quantum information processing operations in constrained quantum systems with finite dimensional Hilbert spaces of states. We demonstrate the approach applied to a spin chain system.