The paper is motivated by the discussion of feasibility of large scale quantum computations which should incorporate both unitarity of quantum dynamics for information bearing degrees of freedom and stability with respect to environmental noise. The minimal thermodynamic cost of a single CNOT gate, which is equivalent to the minimal cost of a quantum measurement of a binary observable is analyzed using a generic quantum model of one bit memory. For this model stability of memory with respect to thermal and quantum noise and the error of readout can be quantified. One obtains the relations between the minimal work which is invested in a measurement or CNOT gate, the error and the stability factor. The basic formula differs from the standard Landauer one and seems to be much more realistic. The results show the fundamental conflict between stability and irreversibility of information processing. This explains the feasibility of classical stable and scalable information processing performed by irreversible gates and suggests impossibility of large scale quantum computations based on unitary gates.
The paper is motivated by the discussion of feasibility of large scale quantum computations which should incorporate both unitarity of quantum dynamics for information bearing degrees of freedom and stability with respect to environmental noise. The minimal thermodynamic cost of a single CNOT gate, which is equivalent to the minimal cost of a quantum measurement of a binary observable is analyzed using a generic quantum model of one bit memory. For this model stability of memory with respect to thermal and quantum noise and the error of readout can be quantified. One obtains the relations between the minimal work which is invested in a measurement or CNOT gate, the error and the stability factor. The basic formula differs from the standard Landauer one and seems to be much more realistic. The results show the fundamental conflict between stability and irreversibility of information processing. This explains the feasibility of classical stable and scalable information processing performed by irreversible gates and suggests impossibility of large scale quantum computations based on unitary gates.
The quantum measurement problem is revisited and discussed in terms of a new solvable measurement model which basic ingredient is the quantum model of a controlled single-bit memory. The structure of this model involving strongly coupled spin and quantum harmonic oscillator allows to define stable pointer states as well-separated Gaussian states of the quantum oscillator and analyze the transition from quantum to classical regime. The relations between accuracy of measurement, stability of pointer states, effective temperature of joint thermal and quantum noise and minimal work needed to perform the bit-flip are derived. They differ from those based on the Landauer principle and are used to analyze thermodynamic efficiency of quantum Szilard engine and imply more realistic estimations of minimal amount of work needed to perform long computations.
This chapter makes us walk a few preliminary, but decisive, steps towards quantum information theory (QIT), which will be the focus of the rest of this book. Here, we shall remain in the classical world, yet getting a hint that it is possible to think of a different world where computations may be reversible, namely, without any loss of information. One key realization through this paradigm shift is that “information is physical.” As we shall see, such a nonintuitive and striking conclusion actually results from the age-long paradox of Maxwell's demon in thermodynamics, which eventually found an elegant conclusion in Landauer's principle. This principle states that the erasure of a single bit of information requires one to provide an energy that is proportional to log 2, which, as we know from Shannon's theory, is the measure of information and also the entropy of a two-level system with a uniformly distributed source. This consideration brings up the issue of irreversible computation. Logic gates, used at the heart of the CPU in modern computers, are based on such computation irreversibility. I shall then describe the computers' von Newman's architecture, the intimate workings of the ALU processing network, and the elementary logic gates on which the ALU is based. This will also provide some basics of Boolean logic, expanding on Chapter 1, which is the key to the following logic-gate concepts.
It is argued using a Gedankenexperiment that a scalable quantum memory could be used as a perpetuum mobile of the second kind and hence cannot be realized in nature. The reasoning is based on the assumption that the Landauer's principle for measurements is a consequence of the second law of thermodynamics and not an independent postulate. This implies a modification of the Landauer's principle when applied for discrimination of equilibrium (metastable) states. While identification of the metastable state can be done at infinitesimally low cost, any change of such state involves dissipation of energy proportional to its stability factor.
A thermodynamically irreversible process generates entropy and dissipates heat. The processing of bits of information can itself at times feature irreversible acts, such as the deletion of a bit, which as demonstrated by Landauer and expanded on by Bennett leads to heat exchange with an environment. This foundationalrelationship between thermodynamic and logical irreversibility invites us to examine this idea in a more general context.Within the domain of quantum information processing, quantum physics mathematically furnishes our information source with a Hilbert space allowing for more general operations. This leads to sources of thermodynamics which were not present in the domain of bits. The generation of non-classical resources such as entanglement towards a computational goal and the accurate autonomous control of a quantum system to carry out a computation both come at a thermodynamic cost. Beyond this, quantum systems are fragile and interaction with their environments leads to error whose impact and mitigation has thermodynamic repercussions.Throughout this project we will explore these connections from different perspectives. Firstly a foundational standpoint, to understand what limitations thermodynamics places on the quantum computations we can carry out and their complexity. Secondly from a pragmatic standpoint, to understand the role ofthermodynamics in the scaling of fault-tolerant quantum computation. In doing so developing a more complete picture on the inherent thermodynamics of quantum information processing and its many different facets.
The potential for scalable quantum computing depends on the viability of fault tolerance and quantum error correction, by which the entropy of environmental noise is removed during a quantum computation to maintain the physical reversibility of the computer’s logical qubits. However, the theory underlying quantum error correction applies a linguistic double standard to the words “noise” and “measurement” by treating environmental interactions during a quantum computation as inherently reversible, and environmental interactions at the end of a quantum computation as irreversible measurements. Specifically, quantum error correction theory models noise as interactions that are uncorrelated or that result in correlations that decay in space and/or time, thus embedding no permanent information to the environment. I challenge this assumption both on logical grounds and by discussing a hypothetical quantum computer based on “position qubits.” The technological difficulties of producing a useful scalable position-qubit quantum computer parallel the overwhelming difficulties in performing a double-slit interference experiment on an object comprising a million to a billion fermions.
In practice, quantum systems are not completely isolated from their environment and the resulting system-environment interaction can lead to information leakage from the system. As a result, if a quantum system is to be used for storing or manipulating information, one would like to characterize these environmental noise effects. Such a characterization affords one the ability to design robust methods for preserving the information contained in the system. Unfortunately, completely characterizing the noise in a realistic amount of time is impossible for even moderately large systems. \n \nIn this thesis we discuss methods and diagnostics for partially characterizing quantum noise processes that are especially useful in quantum information and computation. We present a randomized benchmarking protocol that provides a scalable method for determining important properties of the noise affecting the set of gates used on a quantum information processor. We also prove various properties of the quantum gate fidelity, which is a useful state-dependent measure of the distance between two quantum operations, and an important diagnostic of the noise affecting a quantum process. Some non-intuitive generic features of quantum operations acting on large-dimensional quantum systems are also presented.