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New Approaches on Source Coding for Quantum Stochastic Sources and Implementation of Quantum Fanout Gate

Rabins Wosti

Published Oct 1, 2025• 0 citations
OpenAlex

Abstract

The accurate computation of advanced quantum algorithms like Shor’s integer factorization, quantum phase estimation (QPE), and the quantum Fourier transform (QFT) requires quantum circuits of considerable size and depth. It is difficult to achieve reliable computation with deep quantum circuits due to the limited coherence times of the current noisy quantum devices. The quantum fanout gate is known to be a powerful primitive for reducing the depth of many quantum circuits (Høyer and Špalek 2003; Gottesman and Chuang 1999). Shallow or constant-depth quantum circuits are desirable for both near-term and fault-tolerant quantum computations as they reduce noise and allow faster execution of quantum algorithms, potentially skirting the effects of short coherence times. In this work, we show new approaches towards implementation of quantum fanout gate. In particular, we show that by analogue time-evolving the quantum systems according to two well-studied Hamiltonians, namely, quantum Ising and quantum Heisenberg models, we can implement the quantum fanout operator using constantly many layers of digital quantum gates. Important foundations to the area of quantum encoding were provided by Schumacher who proved the quantum analog of Shannon’s noiseless coding theorem for an independent and identically distributed (i.i.d.) quantum source, (Schumacher 1995). In this work, we show a lossless, variable-length block encoding scheme of quantum information emitted from a completely general stochastic quantum source, and it is encoded into the Fock space. While doing so, we extend the notion of uniquely decodable (or completely lossless) quantum codes to be used for quantum block data compression. As our main result, for a fixed mlmany pure states emitted by a given quantum stochastic source, we derive the optimal lower bound of the average codeword length over a subset of uniquely decodable quantum codes called “special block codes,” which are applied to encode the pure states of m many blocks each of block size l. Additionally, we show that for quantum stationary sources in particular, the optimal lower bound of the average codeword length per symbol computed over a subset of special block codes called “constrained special block codes” equals the von Neumann entropy rate of the source for an asymptotically long block size.

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The accurate computation of advanced quantum algorithms like Shor’s integer factorization, quantum phase estimation (QPE), and the quantum Fourier transform (QFT) requires quantum circuits of considerable size and depth. It is difficult to achieve reliable computation with deep quantum circuits due to the limited coherence times of the current noisy quantum devices. The quantum fanout gate is known to be a powerful primitive for reducing the depth of many quantum circuits (Høyer and Špalek 2003; Gottesman and Chuang 1999). Shallow or constant-depth quantum circuits are desirable for both near-term and fault-tolerant quantum computations as they reduce noise and allow faster execution of quantum algorithms, potentially skirting the effects of short coherence times. In this work, we show new approaches towards implementation of quantum fanout gate. In particular, we show that by analogue time-evolving the quantum systems according to two well-studied Hamiltonians, namely, quantum Ising and quantum Heisenberg models, we can implement the quantum fanout operator using constantly many layers of digital quantum gates. Important foundations to the area of quantum encoding were provided by Schumacher who proved the quantum analog of Shannon’s noiseless coding theorem for an independent and identically distributed (i.i.d.) quantum source, (Schumacher 1995). In this work, we show a lossless, variable-length block encoding scheme of quantum information emitted from a completely general stochastic quantum source, and it is encoded into the Fock space. While doing so, we extend the notion of uniquely decodable (or completely lossless) quantum codes to be used for quantum block data compression. As our main result, for a fixed mlmany pure states emitted by a given quantum stochastic source, we derive the optimal lower bound of the average codeword length over a subset of uniquely decodable quantum codes called “special block codes,” which are applied to encode the pure states of m many blocks each of block size l. Additionally, we show that for quantum stationary sources in particular, the optimal lower bound of the average codeword length per symbol computed over a subset of special block codes called “constrained special block codes” equals the von Neumann entropy rate of the source for an asymptotically long block size.
Consider a general quantum stochastic source that emits at discrete time steps quantum pure states which are chosen from a finite alphabet according to some probability distribution which may depend on the whole history. Also, fix two positive integers mmm and lll. We encode any tensor product of mmm many states emitted by the quantum stochastic source by breaking the tensor product into mmm many blocks where each block has length lll, and considering sequences of mmm many isometries so that each isometry encodes one of these blocks into the Fock space, followed by the concatenation of their images. We only consider certain sequences of such isometries that we call \lq\lq special block codes\rq\rq\ in order to ensure that the string of encoded states is uniquely decodable. We compute the minimum average codeword length of these encodings which depends on the quantum source and the integers mmm, lll, among all possible special block codes. Our result extends the result of [Bellomo, Bosyk, Holik and Zozor, Scientific Reports 7.1 (2017): 14765] where the minimum was computed for one block, i.e.\ for m=1m=1m=1. Lastly, we give a simplified non-adaptive compression technique based on constrained special block codes for general quantum stochastic sources. For quantum stationary sources in particular, we show that the minimum average codeword length per symbol computed over all constrained special block codes is equal to the von-Neumann entropy rate of the source for an asymptotically long block size.
Consider a general quantum stochastic source that emits at discrete time steps quantum pure states which are chosen from a finite alphabet according to some probability distribution which may depend on the whole history. Also, fix two positive integersmmmand l. We encode any tensor product of ml many states emitted by the given quantum source by breaking it intommmmany blocks where each block has length l, and considering sequences ofmmmmany isometries so that each isometry encodes one of these blocks into the Fock space, followed by the concatenation of their images. We only consider certain sequences of such isometries that we call “special block codes” in order to ensure that the string of encoded states is uniquely decodable. We compute the minimum average codeword length of these encodings which depends on the quantum source and the integers m, l, among all possible special block codes. Our result extends the result of [Bellomo, Bosyk, Holik and Zozor, Scientific Reports 7.1 (2017): 14765] where the minimum was computed for one block, i.e. for m = 1.
After K. Bostro/spl uml/m and T. Felbinger observed that lossless quantum data compression does not exist unless decoders know the lengths of codewords, they introduced a classical noiseless channel to inform the decoder of a quantum source about the lengths of codewords. In this paper we analyze their codes and present: 1) a sufficient and necessary condition for the existence of such codes for given lists of lengths of codes; 2) a characterization of the optimal compression rate for their codes. However our main contribution is a more efficient way to use the classical channel. We propose a more general coding scheme. It turned out that the optimal compression can always be achieved by a code obtained by this scheme. A von Neumann entropy lower bound to rates of our codes and a necessary and sufficient condition to achieve the bound are obtained. The gap between this lower bound and the compression rates is also well analyzed. For a special family of quantum sources we provide a sharper lower bound in terms of Shannon entropy. Finally, we propose some problems for further research.
Abstract Quantum noise in real-world devices poses a significant challenge in achieving practical quantum advantage, since accurately compiled and executed circuits are typically deep and highly susceptible to decoherence. To facilitate the implementation of complex quantum algorithms on noisy hardware, we propose an approximate method for compiling target quantum circuits into brick-wall layouts. This new circuit design consists of two-qubit CNOT gates that can be directly implemented on real quantum computers, in conjunction with optimized one-qubit gates, to approximate the essential dynamics of the original circuit while significantly reducing its depth. Our approach is evaluated through numerical simulations of time-evolution circuits for the critical Ising model, quantum Fourier transformation, and Haar-random quantum circuits, as well as experiments on IBM quantum platforms. By accounting for compilation error and circuit noise, we demonstrate that time evolution and quantum Fourier transformation circuits achieve high compression rates, while random quantum circuits are less compressible. The degree of compression is related to the rate of entanglement accumulation in the target circuit. In particular, experiments on IBM platforms achieve a compression rate of 12.512.512.5 for N=12N=12N=12, significantly extending the application of current quantum devices. Furthermore, large-scale numerical simulations for system sizes up to N=30N=30N=30 reveal that the optimal depth dmaxd_{\rm max}dmax​ to achieve maximal overall fidelity is independent of system size NNN, suggesting the scalability of our method for large quantum devices in terms of quantum resources.
We study the problem of compressing a block of symbols (a block quantum state) emitted by a memoryless quantum Bernoulli source. We present a simple-to-implement quantum algorithm for projecting, with high probability, the block quantum state onto the typical subspace spanned by the lending eigenstates of its density matrix. We propose a fixed-rate quantum Shannon-Fano code to compress the projected block quantum state using a per-symbol code rate that is slightly higher than the von Neumann (1955) entropy limit. Finally, we propose quantum arithmetic codes to efficiently implement quantum Shannon-Fano (1948) codes. Our arithmetic encoder and decoder have a cubic circuit and a cubic computational complexity in the block size. Both the encoder and decoder are quantum-mechanical inverses of each other, and constitute an elegant example of reversible quantum computation.
We construct the first known fault-tolerant scheme for injecting states into quantum error-correcting codes with constant space and time overhead. That is, we construct a family of constant-rate quantum error-correcting codes for which a set of bare physical qubits can be injected, i.e. fault-tolerantly encoded, into a code block. Similarly, a code state can be ejected, i.e. fault-tolerantly decoded, back into bare physical qubits. We show that these injection and ejection procedures succeed under circuit-level locally stochastic noise, while incurring just a small constant probability of corrupting each qubit, which is unavoidable for bare physical qubits. We also show how to perform fault-tolerant error correction and code-state preparation under locally stochastic noise. All of our gadgets can be implemented with constant quantum circuit depth (i.e. are single-shot), and with a number of physical qubits growing linearly with the number of logical qubits, assuming the ability to run polynomial-sized noiseless classical circuits on the side. We construct our quantum codes by taking a high-dimensional hypergraph product of classical LDPC codes, which in turn are a simplified version of Spielman's linear-time encodable codes (STOC'95). As our resulting product codes are only resilient to physical errors occurring with non-uniform probabilities across qubits, we then show how to concatenate with inner codes of various sizes to obtain fault-tolerance against uniform noise.