Abstract. We propose definitions of QAC 0, the quantum analog of the classical class AC 0 of constant-depth circuits with AND and OR gates of arbitrary fan-in, and QACC 0 [q], where n-ary MODq gates are also allowed. We show that it is possible to make a ‘cat ’ state on n qubits in constant depth if and only if we can construct a parity or MOD2 gate in constant depth; therefore, any circuit class that can fan out a qubit to n copies in constant depth also includes QACC 0 [2]. In addition, we prove the somewhat surprising result that parity or fanout allows us to construct MODq gates in constant depth for any q, so QACC 0 [2] = QACC 0. Since ACC 0 [p] ̸ = ACC 0 [q] whenever p and q are mutually prime, QACC 0 [2] is strictly more powerful than its classical counterpart, as is QAC 0 when fanout is allowed. 1

This paper investigates the computational power of space-bounded quantum Turing machines. The following facts are proved for space-constructible space bounds s satisfying s(n) = Ω(log n). 1. Any quantum Turing machine (QTM) running in space s can be simulated by an unbounded error probabilistic Turing machine (PTM) running in space O(s). No assumptions on the probability of error or running time for the QTM are required, although it is assumed that all transition amplitudes of the QTM are rational. 2. Any PTM that runs in space s and halts absolutely (i.e., has finite worst-case running time) can be simulated by a QTM running in space O(s). If the PTM operates with bounded error, then the QTM may be taken to operate with bounded error as well, although the QTM may not halt absolutely in this case. In the case of unbounded error, the QTM may be taken to halt absolutely. We therefore have that unbounded error, space O(s) bounded quantum Turing machines and probabilistic Turing machines are equivalent in power, and furthermore that any QTM running in space s can be simulated deterministically in NC 2 (2 s) ⊆ DSPACE(s 2) ∩ DTIME ( 2 O(s)). We also consider quantum analogues of nondeterministic and one-sided error probabilistic spacebounded classes, and prove some simple facts regarding these classes. 1 1

q are distinct primes, QACC[q] is strictly more powerful than its classical counterpart, as is QAC 0 when fanout is allowed. This adds to the growing list of quantum complexity classes which are provably more powerful than their classical counterparts. We also develop techniques for proving upper bounds for QACC in terms of related language classes. We dene classes of languages closely related to QACC[2] and show that restricted versions of them can be simulated by polynomialsize circuits. With further restrictions, language classes related to QACC[2] operators can be simulated by classical threshold circuits of polynomial size and constant depth. Keywords: quantum computation, quantum & circuit complexity, threshold circuit Communicated by : R Cleve & J Watrous 1. Introduction Advances in quantum computation

In the last few years, theoretical study of quantum systems serving as computational devices has achieved tremendous progress. We now have strong theoretical evidence that quantum computers, if built, might be used as a dramatically powerful computational tool, capable of performing tasks which seem intractable for classical computers. This review is about to tell the story of theoretical quantum computation. I left out the developing topic of experimental realizations of the model, and neglected other closely related topics which are quantum information and quantum communication. As a result of narrowing the scope of this paper, I hope it has gained the benefit of being an almost self contained introduction to the exciting field of quantum computation. The review begins with background on theoretical computer science, Turing machines and Boolean circuits. In light of these models, I define quantum computers, and discuss the issue of universal quantum gates. Quantum algorithms, including Shor’s factorization algorithm and Grover’s algorithm for searching databases, are explained. I will devote much attention to understanding what the origins of the quantum computational power are, and what the limits of this power are. Finally, I describe the recent theoretical results which show that quantum computers maintain their complexity power even in the presence of noise, inaccuracies and finite precision. This question cannot be separated from that of quantum complexity, because any realistic model will inevitably be subject to such inaccuracies. I tried to put all results in their context, asking what the implications to other issues in computer science and physics are. In the end of this review I make these connections explicit, discussing the possible implications of quantum computation on fundamental physical questions, such as the transition from quantum to classical physics. 1

We prove a general upper bound on the tradeoff between time and space that suffices for the reversible simulation of irreversible computation. Previously, only simulations using exponential time or quadratic space were known. The tradeoff shows for the first time that we can simultaneously achieve subexponential time and subquadratic space. The boundary values are the exponential time with hardly any extra space required by the Lange-McKenzie-Tapp method and the (log 3)th power time with square space required by the Bennett method. We also give the first general lower bound on the extra storage space required by general reversible simulation. This lower bound is optimal in that it is achieved by some reversible simulations. 1 Introduction Computer power has roughly doubled every 18 months for the last half-century (Moore's law). This increase in power is due primarily to the continuing miniaturization of the elements of which computers are made, resulting in more and more ele...

The need to reverse a computation arises in many contexts---debugging, editor undoing, optimistic concurrency undoing, speculative computation undoing, trace scheduling, exception handling undoing, database recovery, optimistic discrete event simulations, subjunctive computing, etc. The need to analyze a reversed computation arises in the context of static analysis---liveness analysis, strictness analysis, type inference, etc. Traditional means for restoring a computation to a previous state involve checkpoints; checkpoints require time to copy, as well as space to store, the copied material. Traditional reverse abstract interpretation produces relatively poor information due to its inability to guess the previous values of assigned-to variables. We propose an abstract computer model and a programming language---Y-Lisp---whose primitive operations are injective and hence reversible, thus allowing arbitrary undoing without the overheads of checkpointing. Such a computer can be built from reversible conservative logic circuits, with the serendipitous advantage of dissipating far less heat than traditional Boolean AND/OR/NOT circuits. Unlike functional languages, which have one "state " for all times, Y-Lisp has at all times one "state", with unique predecessor and successor states. Compiling into a reversible pseudocode can have benefits even when targeting a traditional computer. Certain optimizations, e.g., update-in-place, and compile-time garbage collection may be more easily performed, because the

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This paper investigates the relative power of spacebounded quantum and classical (probabilistic) computational models. The following relationships are proved. 1. Any probabilistic Turing machine (PTM) which runs in space s and which halts absolutely (i.e. halts with certainty after a finite number of steps) can be simulated in space O(s) by a quantum Turing machine (QTM). If the PTM operates with bounded error, then the QTM may be taken to operate with bounded error as well, although the QTM may not halt absolutely in this case. In the unbounded error case, the QTM may be taken to halt absolutely. 2. Any QTM running in space s can be simulated by an unbounded error PTM running in space O(s). No assumptions on the probability of error or running time for the QTM are required, but it is assumed that all transition amplitudes of the quantum machine are rational. It follows that unbounded error, space O(s) bounded quantum Turing machines and probabilistic Turing machines are equivalent in...

This paper takes up this suggestion to analyze time-space and space-irreversibility tradeoffs. It completely characterizes the realizable pebble configurations of the reversible pebble games (they encode the reachable instantaneous descriptions of a Turing machine reversibly simulating an irreversible computation). As corollary we obtain Bennett's earlier [3] simulaing Group Nr. 8556, and by NWO through NFI Project ALADDIN under Contract number NF 62-376 and NSERC under International Scientific Exchange Award ISE0125663. Affiliations are CWI and the University of Amsterdam. tion and a first proof that this simulation is a space-optimal pebble game. It also introduces irreversible steps and gives a theorem on the tradeoff between the number of allowed irreversible steps and the memory gain in the pebble game. For such a tradeoff the limited irreversible actions have to take place at precise times during the reversible simulation, and cannot be delayed to be executed all together at the end of the computation (as is possible in computations without time or space resource bounds). Finally, in all such reversible simulations it is assumed that the number of steps to be simulated is known in advance and used to construct the simulation (for that number of steps). We show how to reversibly simulate an irreversible computation of unknown computing time, using the same order of magnitude of simulation time. 1.1 Reversible Turing Machines In the standard model of a Turing machine the elementary operations are rules in quadruple format (p; s; a; q) meaning that if the finite control is in state p

We describe a reversible Instruction Set Architecture using recently developed reversible logic design techniques. Such an architecture has the dual advantage of being able to run backwards and of being, in theory, implementable so as to dissipate less than log 2 kT joules per bit operation. We analyze several basic control structures and algorithms on the architecture, showing that, for example, a sorting algorithm need only dissipate O(n log n) bits even though it makes O(n 2 ) comparisons. Keywords reversible computation, entropy, heat dissipation, reversible algorithms, finite automata, computer architecture, retractile cascade, sorting 1 Introduction In [8] we introduced an "electroid" logic for the design of reversible computer architectures. It was at that time a conjecture, based on an observation of Merkle [10], that such a logic could be used to design a computer architecture that was not radically different from conventional ones at the instruction set level, and yet kept...