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...odel of computation, augmented with measurement operations during the computation. For noise-free quantum circuits, these measurement operations can always be delayed until the end of the computation =-=[2]-=-; thus previous definitions of quantum circuits have sometimes only permitted measurement steps at the end, as this model is easier to work with and was believed to be equivalent. It has not yet been ...

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...of a noisy quantum system is in general a probability distribution over pure states, i.e. a mixed state, and not merely a pure state as in the standard model, we use quantum circuits with mixed states=-=[3]-=-. Since noise is a dynamic process which depends on time, the circuit will be divided to levels, or time steps. Unlike Yao[25] we allow qubits to be input and output at different times to the circuit....

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... can be used to obtain a quadratic speed up for certain randomized algorithms | such as 2-SAT. Indeed, all quantum algorithms from the last decade | including Shor's celebrated factorization algorithm=-=[1]-=- and Grover's search algorithm[2] | use only quantum Fourier transforms and classical computation. Is it possible to use other types of unitary transformations to design new quantum algorithms? One co...

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...X) denotes the Shannon entropysof X; H(XjY ), the conditionalShannon entropy of X with respect to the variable Y ; and I(X : Y ), the mutual informationsof the two variables X and Y . We also use H : =-=[0; 1]-=- ! [0; 1] to denote the binary entropy function. We refer the reader to [7] for the definition and properties of these standard concepts from classical information theory. The quantum mechanical analo...

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...ents are taking place at various times during the circuit's computation; such measurements, however, do not occur and are assumed to be simulated in the sense described by Aharonov, Kitaev, and Nisan =-=[2]-=-. For each circuit Q x , some number of the qubits upon which Q x acts are specified as input qubits, and all other qubits are ancilla qubits. The input qubits are assumed to be initialized in some sp...

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...tic computation whose output is ǫ/2 close in ℓ2-norm to |η〉 and whose running time is f(L ′ ,ǫ/2). Since the trace distance between two pure states is bounded from above by the ℓ2distance (see, e.g., =-=[3]-=-), we obtain that the output of the adiabatic computation is also ǫ/2 close in trace distance to |η〉〈η|. In addition, it is easy to check that after we trace out the clock qubits from |η〉, we are left...

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... i 2 H K satisfy trK jihj = trK j ihsj, then there exists U 2 U(K) such that (I U)ji = j i. Proofs of the facts comprising Theorem 1 can be found as follows: 1. see page 430 of [20], 2. and 3. see [1]=-=-=- or [23], 4. see [22], 5. see [15], and 6. see [21]. 2.2 Quantum circuits The computational model upon which quantum interactive proof systems are based is the (acyclic) quantum circuit model. See [1;...

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...es (languages recognized by 1-way deterministic automata). A more powerful generalization of this model is 1-way qfa’s that allow mixed states (defined similarly to quantum circuits with mixed states =-=[1]-=-). Any 1-way dfa can be easily simulated by a 1-way qfa with mixed states. However, all languages recognized by 1-way qfa’s with mixed states (with bounded error) are still regular. The second model i...

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...ons of indistinguishability of these mappings Φx and Ψx. The correct quantum analogue of statistical zero-knowledge requires that ‖Φx −Ψx‖# is negligible, where ‖ · ‖# denotes Kitaev’s “diamond” norm =-=[18, 19, 2]-=-. Informally this implies that no physical process can distinguish Φx and Ψx given a single “blackbox” access to one of the two mappings, including the possibility that the mapping is applied to just ...