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New Journal of Physics

Multi-pulse Fourier codes for bit transmission at the quantum limit

Matteo Rosati

Published Jan 1, 2025• 0 citations
Publisher

Abstract

Abstract Bit-transmission can be enhanced by the use of quantum detection techniques, realizing a joint-detection receiver (JDR) that is able to decode transmitted signals via a collective operation and achieve the Holevo channel capacity. Explicit JDR designs proposed so far employ the Hadamard or Fourier transform to perform a phase-to-intensity translation of the information encoding, effectively falling in the class of on-off-keying (OOK) modulation techniques; they improve over classical decoders but fall short of the Holevo capacity, particularly at large signal mean photon number n ≳ 1 . Here we introduce new families of decoders based on multi-pulse and multi-level codes. We compute the rate of these codes exactly, and provide a comprehensive study of their performance. We show that multi-pulse codes can approach the rate of OOK closely, providing a simplified design for quantum-enhanced communication in the photon-starved regime; furthermore, multi-level codes can approach generalized-OOK strategies with multiple pulse types, thus they can be employed in the larger photon-number regime.

DOI
10.1088/1367-2630/ada4d0

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Abstract Bit-transmission can be enhanced by the use of quantum detection techniques, realizing a joint-detection receiver (JDR) that is able to decode transmitted signals via a collective operation and achieve the Holevo channel capacity. Explicit JDR designs proposed so far employ the Hadamard or Fourier transform to perform a phase-to-intensity translation of the information encoding, effectively falling in the class of on-off-keying (OOK) modulation techniques; they improve over classical decoders but fall short of the Holevo capacity, particularly at large signal mean photon number n ≳ 1 . Here we introduce new families of decoders based on multi-pulse and multi-level codes. We compute the rate of these codes exactly, and provide a comprehensive study of their performance. We show that multi-pulse codes can approach the rate of OOK closely, providing a simplified design for quantum-enhanced communication in the photon-starved regime; furthermore, multi-level codes can approach generalized-OOK strategies with multiple pulse types, thus they can be employed in the larger photon-number regime.
Optical communication technology can be enhanced by using quantum signals to transfer classical bits. This requires the message-carrying signals to interact coherently at the decoder via a joint-detection receiver (JDR). To date, the realization of a JDR using optical technologies has remained elusive: the only explicit design, called a Green–Hadamard receiver (GHR), increases distinguishability at the cost of reducing the code size. We introduce a supervised-learning framework for the systematic discovery of optical JDR designs based on parametrized photonic integrated circuits. We find JDR designs with higher decoding success probability than any single-symbol receiver, including homodyne, Kennedy, and Dolinar. Furthermore, our new receiver families surpass the GHR receiver for mean photon number >0.1, both in terms of code size and decoding probability, paving the way for practical applications of JDR in optical fiber networks and free-space.
For space-based laser communications, when the mean photon number per received optical pulse is much smaller than one, there is a large gap between communications capacity---measured in bits communicated per pulse---achievable with a receiver that detects (converts from optical to electrical domain) each modulated pulse one at a time, versus with the quantum-optimal joint-detection receiver that acts on a long codeword comprised of nnn modulated pulses; an effect often termed superadditive capacity. The action of this receiver cannot be described as the detection of each individual pulse, interspersed with classical feedforward and soft-information post-processing. In this paper, we consider the simplest scenario where a large superadditive capacity is known: a pure-loss channel with a coherent-state binary phase-shift keyed (BPSK) modulation. The two BPSK states can be mapped conceptually to two non-orthogonal states of a single qubit, described by an inner product that is a function of the mean photon number of each BPSK pulse. Using this map, we derive an explicit construction of the quantum circuit of a joint-detection receiver based on a recent idea of belief-propagation with quantum messages (BPQM) [arXiv:1607.04833]. We analyze this scheme rigorously and show that it achieves the quantum limit of minimum average error probability in discriminating 888 (BPSK) codewords of a length-555 binary linear code with a tree factor graph. We quantify its performance improvement over the (Dolinar) receiver that optimally detects one pulse at a time. Our result suggests that a BPQM-receiver might attain the Holevo capacity, the quantum limit of classical communication capacity, of this BPSK-modulated pure-loss channel. This suggests a new application for a small, special-purpose, photonic quantum computer capable of so-called cat-basis universal qubit logic.
Abstract For space-based laser communications, when the mean photon number per received optical pulse is much smaller than one, there is a large gap between communications capacity achievable with a receiver that performs individual pulse-by-pulse detection, and the quantum-optimal “joint-detection receiver” that acts collectively on long codeword-blocks of modulated pulses; an effect often termed “superadditive capacity”. In this paper, we consider the simplest scenario where a large superadditive capacity is known: a pure-loss channel with a coherent-state binary phase-shift keyed (BPSK) modulation. The two BPSK states can be mapped conceptually to two non-orthogonal states of a qubit, described by an inner product that is a function of the mean photon number per pulse. Using this map, we derive an explicit construction of the quantum circuit of a joint-detection receiver based on a recent idea of “belief-propagation with quantum messages” (BPQM). We quantify its performance improvement over the Dolinar receiver that performs optimal pulse-by-pulse detection, which represents the best “classical” approach. We analyze the scheme rigorously and show that it achieves the quantum limit of minimum average error probability in discriminating 8 (BPSK) codewords of a length-5 binary linear code with a tree factor graph. Our result suggests that a BPQM receiver might attain the Holevo capacity of this BPSK-modulated pure-loss channel. Moreover, our receiver circuit provides an alternative proposal for a quantum supremacy experiment, targeted at a specific application that can potentially be implemented on a small, special-purpose, photonic quantum computer capable of performing cat-basis universal qubit logic.
A scheme for transferring classical information over a lossy bosonic channel is proposed by generalizing the proposal presented by Guha [Phys. Rev. Lett. 106, 240502 (2011)]. It employs code words formed by products of coherent states of fixed mean photon number with multiple phases which, through a passive unitary transformation, reduce to a pulse-position modulation code with multiple pulse phases. The maximum information rate achievable with optimal, yet difficult to implement detection schemes is computed and shown to saturate the classical capacity of the channel in the low-energy regime. An easy to implement receiver based on a conditional Dolinar detection scheme is also proposed, finding that while suboptimal, it allows for improvements in an intermediate photon-number regime with respect to previous proposals.
Quantum information theory predicts that communication technology can be enhanced by using quantum signals to transfer classical bits. In order to fulfill this promise, the message-carrying signals must interact coherently at the decoding stage via a joint-detection receiver (JDR), whose realization with optical technologies remains an outstanding open problem to date. We introduce a supervised-learning framework for the systematic discovery of new JDR designs based on parametrized photonic integrated circuits. Our framework relies on the synthesis of a training set comprising quantum codewords and the corresponding classical message label; the codewords are processed by the JDR circuit and, after photo-detection, produce a guess for the label. The circuit parameters are then updated by minimizing a suitable loss function, reaching an optimal JDR design for that specific architecture. We showcase our method with coherent-state codes for the pure-loss bosonic channel, modelling optical-fiber and space communication, with a circuit architecture comprising linear optics, squeezing and threshold photo-detectors. We train JDR circuits for several code families, varying energy and code-size. We discover optical JDR circuit setups for maximum-size codes and small message-length that offer up to a 333-fold enhancement in the bit decoding rate with respect to the optimal single-symbol receiver, and less than 7%7\%7%-away from the theoretically optimal decoder, for which an explicit design is missing to date. Furthermore, the discovered receivers surpass previous JDR designs both in terms of bit decoding and bit transmission rate. Finally, we observe that the best-performing codes are those which can be mapped, via the JDR's optical processing, to modulations with different energy levels on different symbols, making the message symbols more distinguishabile via photo-detection.
The fundamental performance limits and channel capacity of optical communications systems operating over the free space channel will be examined using quantum detection theory. The performance of the optimum quantum receiver for on-off keying (OOK) and optical binary phase shift keying (BPSK) is first examined as a pure state (no noise) problem. The classical capacity of the binary symmetric channel for these two modulation schemes is then evaluated for the optimum quantum receiver by making use of the concept of quantum measurement states. The performance of M-ary pulse position modulation, which requires a product state representation, is evaluated along with the performance of certain 'dense signal sets.' Performance comparisons with classical techniques shows over 5 dB improvement in some cases when quantum detection is employed. As a further application of the quantum detection theory, the capacity of the binary channel with on-off keyed modulation and quantum detection is evaluated, and shown to exceed the capacity obtained with classical photon counting.