Usually, a proof of a theorem contains more knowledge than the mere fact that the theorem is true. For instance, to prove that a graph is Hamiltonian it suffices to exhibit a Hamiltonian tour in it; however, this seems to contain more knowledge than the single bit Hamiltonian/non-Hamiltonian.In this paper a computational complexity theory of the “knowledge” contained in a proof is developed. Zero-knowledge proofs are defined as those proofs that convey no additional knowledge other than the correctness of the proposition in question. Examples of zero-knowledge proof systems are given for the languages of quadratic residuosity and 'quadratic nonresiduosity. These are the first examples of zero-knowledge proofs for languages not known to be efficiently recognizable.

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The knowledge complexity of interactive proof-systems

Introduction Classical computation Quantum computation Solutions Elementary number theory Bibliography Index.

Classical and Quantum Computation

Fault tolerance is the study of reliable computation using unreliable components. With a given noise model, can one still reliably compute? For example, one can run many copies of a classical calculation in parallel, periodically using majority gates to catch and correct faults. Von Neumann showed in 1956 that if each gate fails independently with probability p, flipping its output bit 0 $ 1, then such a fault tolerance scheme still allows for arbitrarily reliable computation provided that p is below some constant threshold (whose value depends on the model details) [10]. In a quantum computer, the basic gates are much more vulnerable to noise than classical transistors – after all, depending on the implementation, they are manipulating single electron spins, photon polarizations, and similarly fragile subatomic particles. It might not be possible to engineer systems with noise rates less than 10 , or perhaps 10 , per gate. Additionally, the phenomenon of entanglement makes quantum systems inherently fragile. For example, in Schrödinger’s cat state – an equal superposition between a living cat and a dead cat, often idealized as 1= p 2.j0ni C j1ni/ – an interaction with just one quantum bit (“qubit”) can collapse, or decohere, the entire system. Fault tolerance techniques will therefore be essential for achieving the considerable potential of quantum computers. Practical fault tolerance techniques will need to control high noise rates and do so with low overhead, since qubits are expensive. Quantum systems are continuous, not discrete, so there are many possible noise models. However, the essential features of quantum noise for fault tolerance results can be captured by a simple discrete model similar to the one Von Neumann used. The main difference is that, in addition to bit-flip X errors which swap 0 and 1, there can also be phase-flip Z errors which swap jCi 1=p2.j0i C j1i/ and j i 1=p2.j0i j1i/ (Fig. 1). A noisy gate is modeled as a perfect gate followed by independent introduction of X, Z, or Y (which is both X and Z) errors with respective probabilities pX; pZ; pY. One popular model is independent depolarizing noise .pX D pZ D pY p=3/; a depolarized qubit is completely randomized. Faulty measurements and preparations of single-qubit states must additionally be modeled, and there can be memory noise on resting qubits. It is often assumed that measurement results can

Fault-Tolerant Quantum Computation.

Quantum computers can execute algorithms that dramatically outperform classical computation. As the best-known example, Shor discovered an efficient quantum algorithm for factoring integers, whereas factoring appears to be difficult for classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum computation, and such algorithms motivate the formidable task of building a large-scale quantum computer. This article reviews the current state of quantum algorithms, focusing on algorithms with superpolynomial speedup over classical computation and, in particular, on problems with an algebraic flavor.

Quantum algorithms for algebraic problems

It is shown that BPP can be simulated in subexponential time for infinitely many input lengths unless exponential time collapses to the second level of the polynomial-time hierarchy, has polynomial-size circuits, and has publishable proofs (EXPTIME=MA). It is also shown that BPP is contained in subexponential time unless exponential time has publishable proofs for infinitely many input lengths. In addition, it is shown that BPP can be simulated in subexponential time for infinitely many input lengths unless there exist unary languages in MA/P. The proofs are based on the recent characterization of the power of multiprover interactive protocols and on random self-reducibility via low degree polynomials. They exhibit an interplay between Boolean circuit simulation, interactive proofs and classical complexity classes. An important feature of this proof is that it does not relativize.<<ETX>>

BPP has Subexponential Time Simulation unless EXPTIME has Pubishable Proofs.

This paper introduces quantum “multiple-Merlin”-Arthur proof systems in which Arthur uses multiple quantum proofs unentangled with each other for his verification. Although classical multi-proof systems are obviously equivalent to classical single-proof systems, it is unclear whether quantum multi-proof systems collapse to quantum single-proof systems. This paper presents a necessary and sufficient condition under which the number of quantum proofs is reducible to two. It is also proved that using multiple quantum proofs does not increase the power of quantum Merlin-Arthur proof systems in the case of perfect soundness, and that there is a relativized world in which co-NP (actually co-UP) does not have quantum Merlin-Arthur proof systems even with multiple quantum proofs.

Quantum Merlin-Arthur Proof Systems: Are Multiple Merlins More Helpful to Arthur?

Theory of one-tape linear-time Turing machines

This paper introduces quantum ``multiple-Merlin''-Arthur proof systems in which Arthur receives multiple quantum proofs that are unentangled with each other. Although classical multi-proof systems are obviously equivalent to classical single-proof systems (i.e., usual Merlin-Arthur proof systems), it is unclear whether or not quantum multi-proof systems collapse to quantum single-proof systems (i.e., usual quantum Merlin-Arthur proof systems). This paper presents a necessary and sufficient condition under which the number of quantum proofs is reducible to two. It is also proved that, in the case of perfect soundness, using multiple quantum proofs does not increase the power of quantum Merlin-Arthur proof systems.

Polynomial time quantum computation with advice

Advice is supplementary information that enhances the computational power of an underlying computation. This paper focuses on advice that is given in the form of a pure quantum state and examines the influence of such advice on the behaviors of an underlying polynomial-time quantum computation with bounded-error probability.

Tomoyuki Yamakami

Papers

Quantum Merlin-Arthur Proof Systems: Are Multiple Merlins More Helpful to Arthur?

Theory of one-tape linear-time Turing machines

Quantum Merlin-Arthur Proof Systems: Are Multiple Merlins More Helpful to Arthur?

Polynomial time quantum computation with advice

Polynomial time quantum computation with advice