Quantum operations provide a general description of the state changes allowed by quantum mechanics. The reversal of quantum operations is important for quantum error-correcting codes, teleportation, and reversing quantum measurements. We derive information-theoretic conditions and equivalent algebraic conditions that are necessary and sufficient for a general quantum operation to be reversible. We analyze the thermodynamic cost of error correction and show that error correction can be regarded as a kind of ``Maxwell demon,'' for which there is an entropy cost associated with information obtained from measurements performed during error correction. A prescription for thermodynamically efficient error correction is given.
I give a pedagogical account of Shor's nine-bit code for correcting arbitrary errors on single qubits, and I review work that determines when it is possible to maintain quantum coherence by reversing the deleterious effects of open-system quantum dynamics. The review provides an opportunity to introduce an efficient formalism for handling superoperators. I present and prove some bounds on entanglement fidelity, which might prove useful in analyses of approximate error correction.
Twenty-one characterizations of reversible quantum channels
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学科学重点实验室）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；School of Mathematics and Statistics, Shaanxi Normal University（陕西师范大学数学与统计学院）
A quantum measurement is logically reversible if the premeasurement density operator of the measured system can be calculated from the postmeasurement density operator and from the outcome of the measurement. This paper analyzes why many quantum measurements are logically irreversible, shows how to make them logically reversible, and discusses reversing measurement that returns the postmeasurement state to the premeasurement state by another measurement (physical reversibility). Reversing measurement and unitarily reversible quantum operation are compared from the viewpoint of error correction…
We show that any unitary transformation performed on the quantum state of a closed quantum system, describes an inner, reversible, generalized quantum measurement. We also show that under some specific conditions it is possible to perform a unitary transformation on the state of the closed quantum system by means of a collection of generalized measurement operators. In particular, given a complete set of orthogonal projectors, it is possible to implement a reversible quantum measurement that preserves the probabilities. In this context, we introduce the concept of "Truth-Observable", which is …
Landauer's principle asserts that any computation has an unavoidable energy cost that grows proportionally to its degree of logical irreversibility. But even a logically reversible operation, when run on a physical processor that operates on different energy levels, requires energy. Here we quantify this energy requirement, providing upper and lower bounds that coincide up to a constant factor. We derive these bounds from a general quantum resource-theoretic argument, which implies that the initial resource requirement for implementing a unitary operation within an error~$ε$ grows like $1/\sqr…
It has been shown (Arxiv: quant-ph/0507236) that a universal quantum computer could be powerful enough to solve efficiently the quantum search problem, and the reversible and unitary halting protocol based on the state-locking pulse field is the key component to construct the efficient quantum search processes, while the state-locking pulse field is the key component to generate the reversible and unitary halting protocol. In this paper the reversible and unitary halting protocol and the generalized state-locking pulse field have been extensively investigated theoretically. The basic principle…
Reversible logic circuits have been historically motivated by theoretical research in low-power electronics as well as practical improvement of bit-manipulation transforms in cryptography and computer graphics. Recently, reversible circuits have attracted interest as components of quantum algorithms, as well as in photonic and nano-computing technologies where some switching devices offer no signal gain. Research in generating reversible logic distinguishes between circuit synthesis, post-synthesis optimization, and technology mapping. In this survey, we review algorithmic paradigms --- search…
In previous work and motivated by a theoretical discussion on physical realizations, a new quantum gate library (the NCV-v1 library) for electronic design automation of quantum circuits has been proposed. Here, qudits instead of qubits are assumed, i.e. a basic building block which does not rely on a two level quantum system but a (multiple-valued) d-level quantum system is assumed. However, the descriptions on the foundation of this new library remained brief. This technical report provides an extended description of the applied ideas and concepts.
In quantum computing the decoherence time of the qubits determines the computation time available and this time is very limited when using current hardware. In this paper we minimize the execution time (the depth) for a class of circuits referred to as linear reversible circuits, which has many applications in quantum computing (e.g., stabilizer circuits, CNOT+T circuits, etc.). We propose a practical formulation of a divide and conquer algorithm that produces quantum circuits that are twice as shallow as those produced by existing algorithms. We improve the theoretical upper bound of the dept…
The Hidden Weighted Bit function plays an important role in the study of classical models of computation. A common belief is that this function is exponentially hard for the implementation by reversible ancilla-free circuits, even though introducing a small number of ancillae allows a very efficient implementation. In this paper, we refute the exponential hardness conjecture by developing a polynomial-size reversible ancilla-free circuit computing the Hidden Weighted Bit function. Our circuit has size $O(n^{6.42})$, where $n$ is the number of input bits. We also show that the Hidden Weighted B…
We show how to reconstruct a process theory of local systems starting from a global theory of reversible processes on a single global system, by using the purification principle. In such a process theory, local systems are not given, but rather `emerge' as the global system is decomposed into subsystems. Local systems thus have specific identities and their composition is naturally limited by structural constraints, a behaviour which we formalise by defining symmetric partially-monoidal categories. We reconstruct quantum theory from the global theories of unitary groups acting on projective Hi…
Operator quantum error-correction is a technique for robustly storing quantum information in the presence of noise. It generalizes the standard theory of quantum error-correction, and provides a unified framework for topics such as quantum error-correction, decoherence-free subspaces, and noiseless subsystems. This paper develops (a) easily applied algebraic and information-theoretic conditions which characterize when operator quantum error-correction is feasible; (b) a representation theorem for a class of noise processes which can be corrected using operator quantum error-correction; and (c)…
We propose a quantum generalisation of a classical neural network. The classical neurons are firstly rendered reversible by adding ancillary bits. Then they are generalised to being quantum reversible, i.e.\ unitary. (The classical networks we generalise are called feedforward, and have step-function activation functions.) The quantum network can be trained efficiently using gradient descent on a cost function to perform quantum generalisations of classical tasks. We demonstrate numerically that it can: (i) compress quantum states onto a minimal number of qubits, creating a quantum autoencoder…
We establish that, in an appropriate limit, qubits of communication should be regarded as composite resources, decomposing cleanly into independent correlation and transmission components. Because qubits of communication can establish ebits of entanglement, qubits are more powerful resources than ebits. We identify a new communications resource, the zero-bit, which is precisely half the gap between them, replacing classical bits by zero-bits makes teleportation asymptotically reversible. The decomposition of a qubit into an ebit and two zero-bits has wide-ranging consequences including applica…
After more than a century since its birth, Quantum Theory still eludes our understanding. If asked to describe it, we have to resort to abstract and ad hoc principles about complex Hilbert spaces. How is it possible that a fundamental physical theory cannot be described using the ordinary language of Physics? Here we offer a contribution to the problem from the angle of Quantum Information, providing a short non-technical presentation of a recent derivation of Quantum Theory from information-theoretic principles. The broad picture emerging from the principles is that Quantum Theory is the only…
