A. C.-C. Yao, "Quantum circuit complexity", Proc. of the 34th FOCS, pp. 352--361, 1993.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....this, our results are derived from first principles, using a linear algebraic technique that has its roots in the work of Nayak [16] In order to prove Theorem 1. 1, we give a new characterisation of the joint state at the end of a quantum protocol that complements the characterisation due to Yao [20]. It greatly clarifies the role of shared entanglement in communication, and we expect that it will further enhance our conceptual understanding of quantum communication. Putting Theorem 1.1 together with a reduction due to Cleve et al. 8] we get a new lower bound of 2 (n 2 log 1 2# ) for the ....

....general encoding as well, which is the subject of Section 3.2. Building on the insight gained from the study of quantum encoding, we extend our results to the case of interactive communication in Section 4. 2. PRELIMINARIES 2. 1 The communication model In the quantum communication model of Yao [20], two parties Alice and Bob hold qubits. When the game starts Alice holds a superposition x# and Bob holds y#, representing the input to the two players. The initial joint state is thus x#A# y#B , where a subscript indicates the player holding that set of qubits. Furthermore each player ....

A. C.-C. Yao. Quantum circuit complexity. In Proceedings of the 34th Annual IEEE Symposium on Foundations of Computer Science, pages 352--361, 1993.

....described a universal simulator for QTMs with exponential overhead. Bernstein and Vazirani [3] were able to construct a universal QTM with only polynomial overhead. Other quantum computational models were also studied recently. Deutsch [6] has defined the model of quantum circuits, and later Yao [19] has shown that QTMs working in polynomial time can be simulated by polynomial size quantum circuits. Physicists were also interested in quantum cellular automata: Biafore [4] considered the problem of synchronization, Margolus [14] described space periodic quantum cellular automata and Lloyd [12, ....

A. Yao, Quantum circuit complexity, Proceedings of the 34th IEEE Symposium on Foundations of Computer Science, (1993), pp. 352--361. 15

....that logic synthesis is a mature field in our community. However, this is only true for conventional ANDOR NOT based circuits or LSI s; new ideas must be needed if we face technology innovations. The main purpose of this paper is to introduce logic synthesis for quantum Boolean circuits [12] (QBCs for short) The key difference between conventional circuits and quantum ones is their base family of logic gates, for the latter of which many people agree that Control Not (CNOT) type logic gates will be a single possibility. Another (even more important) feature of QBCs is its severe ....

....rule set is complete, which means we can transform any QBC into any of its equivalent ones by applying these rules. We also make some concrete suggestions on how to use these transformation rules to simplify QBCs. It should be noted that quantum algorithms are often described by using QBCs [12]. Designing a good QBC is thus plays a key role to the successful implementation of a quantum algorithm. A little surprisingly, however, relatively small attention has been paid for the design methodology of QBCs [1, 9] 1] shows that any unitary transformation can be broken down into a ....

A. Yao. Quantum circuit complexity. In Proc. 34th Annual IEEE Symposium on Foudations of Computer Science, pages 352--361, 1993.

....there is a PAC learning algorithm for C which has poly(n; sample complexity, and we say that C is eciently PAC learnable if there is a PAC learning algorithm for C which runs in poly(n; time. 2. 3 Quantum Computation Detailed descriptions of the quantum computation model can be found in [6, 14, 31]; here we outline only the basics using the terminology of quantum networks as presented in [4] A quantum network N is a quantum circuit (over some standard basis augmented with one oracle gate) which acts on an m bit quantum register; the computational basis states of this register are the 2 ....

A.C. Yao. Quantum circuit complexity, in \Proc. 34th Symp. on Found. of Comp. Sci." (1993), 352-361.

....on the complexity of the inner product IP n , and [BFS86] also contained an n) lower bound for DISJ n . The latter bound was improved to the optimal n) in [KS92] and their proof was further simpli ed in [Raz92] The model of quantum communication complexity was also introduced by Yao [Yao93] Suppose that Alice and Bob can employ the laws of quantum mechanics and are allowed to exchange qubits instead of classical bits. Can it help them to reduce the amount of communication Buhrman, Cleve and Wigderson [BCW98] observed that the rank lower bound for deterministic protocols extends ....

....and Bob share an unlimited number of entangled EPRpairs before the communication even begins) The question about the complexity of protocols that allow a small error is by far more interesting. As far as lower bounds are concerned, Kremer [Kre95] based upon some ideas from the seminal paper [Yao93] proved an n) lower bound for IP n . This result was extended to the model with prior entanglement in [CDNT98] Klauck [Kla01] looked at the threshold predicates (D(s) s ) and exact predicates (D(s) s = and proved an = log ) bound in both cases (without entanglement) The ....

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A. Yao. Quantum circuit complexity. In Proceedings of the 34th IEEE Symposium on Foundations of Computer Science, pages 352-361, 1993. 20

.... lie in quantum mechanical computing, several classical restricted models of computation have been rede ned as quantum computing systems, e.g. quantum nite automata, quantum decision trees (aka black box computations) quantum formulae, and quantum communication protocols (see [24] 15] [38]) The nice applications of communication complexity results to lower bound proofs are motivation to study this model also in the quantum case (introduction to (classical) communication complexity can be found in [27] and [19] This paper has appeared in the Proceedings of the 32nd ACM ....

....the quantum case (introduction to (classical) communication complexity can be found in [27] and [19] This paper has appeared in the Proceedings of the 32nd ACM Symposium on Theory of Computing, 2000. an overview on quantum communication complexity in [37] In a quantum protocol (as de ned in [38]) the players exchange qubits rather than bits. Another scenario, where the players may also possess some (input independent) qubits that are entangled with the other players qubits is proposed in [11] and [12] Due to the superdense coding technique of [5] in this model 2 classical bits can be ....

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A.C. Yao. Quantum Circuit Complexity. 34th Symp. Found. of Computer Science, pp. 352-361, 1993. 19

....lower bound results rely heavily on quantum information theory. The necessary background is provided in Section II B, along with the associated notation. See also [22] for a thorough introduction into the eld. 7 A. The communication complexity model In the quantum communication complexity model [23], two parties Alice and Bob hold qubits. When the game starts Alice holds a superposition jxi and Bob holds jyi (representing the input to the two players) and so the initial joint state is simply jyi. Furthermore each player has an arbitrarily large supply of private qubits in some xed basis ....

....overall communication is at most kc, and the number of rounds used is always the worst case number of rounds. In the example above we get a 2c communication complexity. This restriction is also implicit in Yao s de nition of quantum communication complexity using interacting quantum circuits [23]. 9 B. Information theory background The quantum mechanical analogue of a random variable is a probability distribution over superpositions, also called a mixed state. For the mixed state X = fp i ; j i ig, where j i i has probability p i , the density matrix is de ned as X = i p i j i ....

A.C.-C. Yao, \Quantum circuit complexity," in Proceedings of the 34th Annual IEEE Symposium on Foundations of Computer Science, 1993, pp. 352-361.

....quantum computation becomes practical. Institutes of Physics and Computer science, The Hebrew University, Jerusalem, Israel, E mail: doria cs.huji.ac.il y Institute of Computer science, The Hebrew University, Jerusalem, Israel, E mail: benor cs.huji.ac. il 1 Introduction Quantum computation[22, 23, 74] is believed to be more powerful than classical computation, due to oracle results[64, 9] and Shor s algorithm[61] It is yet unclear whether and how quantum computers will be physically realizable, 49, 25, 18] but as any physical system, they in principle will be subjected to noise, such as ....

....quantum computation [3] such that errors and error corrections can be described entirely inside this model. In this paper we are able to prove the main result due to working in such a formal framework. Sequential quantum computation can not be noise resistant[2] so we work with quantum circuits[23, 74]. As the state of a noisy quantum system is in general a probability distribution over pure states, i.e. a mixed state[59] 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 ....

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Yao A C-C, Quantum circuit complexity, in 33th Annual Symposium on Foundations of Computer Science(FOCS), (1993) pp. 352-361

....rate lower than some constant j 2 . The transition from exponential cost to polynomial cost happens in a short range of decoherence rates. We use computer experiments to exhibit the phase transitions in various quantum circuits. 1 Introduction Quantum Turing Machines[8, 4] and Quantum Circuits[7, 19] challenge the so called polynomial Church thesis which asserts that randomized Turing machines can simulate with polynomial slowdown any computational device. In particular Shor s quantum factoring algorithm[16] provides within this theoretical framework, an efficient solution, to a problem ....

....zero. Given a density matrix ae of n qubits, the reduced density matrix of a subsystem,A, of, say, m qubits is defined as an average over the states of the other qubits: aej A (i; j) P 2 n Gammam k=1 ae(ik; jk) 2. 2 Quantum circuits with mixed states We describe the model of quantum circuits[7, 19], with mixed states[2] A quantum unitary gate of order k is a complex unitary matrix of size 2 k Theta 2 k . A density matrix ae will transform by the gate to g ffi ae = U ae U y , where U is the extension of U . Using density matrices one can also define a non unitary gate: A ....

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A. Yao. Quantum circuit complexity. In 34th Annual Symposium on Foundations of Computer Science, pages 352--361, 1993. 10

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A. C.-C. Yao, "Quantum circuit complexity", Proc. of the 34th FOCS, pp. 352--361, 1993.

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A.C. Yao, "Quantum circuit complexity", Proc. of the 34th Ann. IEEE Symp. on Foundations of Computer Science, 1993, pp. 352--361.

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A.C. Yao, "Quantum circuit complexity", Proc. of the 34th Ann. IEEE Symp. on Foundations of Computer Science, 1993, pp. 352--361.

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A. C.-C. Yao, "Quantum circuit complexity", Proc. of the 34th FOCS, pp. 352--361, 1993.

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A.C.-C. Yao. Quantum circuit complexity. In Proceedings of the 34th IEEE Symposium on Foundations of Computer Science, pages 352--361, 1993. 10

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A. C.-C. Yao. Quantum circuit complexity. In Proceedings of the 34th IEEE Symposium on Foundations of Computer Science, pages 352--361, 1993.

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A.C.-C. Yao, "Quantum circuit complexity," in Proceedings of the 34th Annual IEEE Symposium on Foundations of Computer Science, 1993, pp. 352--361.

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A. C. Yao. Quantum circuit complexity. In 34th Annual Symposium on Foundations of Computer Science, pages 352-361, 1993.

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A. Yao. Quantum circuit complexity. In Proceedings of the 34th Annual Symposium on Foundations of Computer Science, pages 352--361, 1993.

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A. Yao. Quantum circuit complexity. In Proceedings of the 34th Annual Symposium on Foundations of Computer Science, pages 352--361, 1993.

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A. C. Yao. Quantum circuit complexity. In IEEE Symposium on Foundations of Computer Science (FOCS), pages 352--361, 1993.

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A. C-C. Yao. Quantum circuit complexity. In Proceedings of 34th IEEE FOCS, pages 352-360, 1993. 21

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A.C.-C. Yao, "Quantum circuit complexity," in Proceedings of the 34th Annual IEEE Symposium on Foundations of Computer Science, 1993, pp. 352--361.

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Yao A C-C, Quantum circuit complexity, in 33th Annual Symposium on Foundations of Computer Science(FOCS), (1993) pp. 352--361

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A. Yao. Quantum Circuit Complexity. In FOCS'93, pp. 351-- 361, 1993. A Shor's Algorithm and Simon's Algorithm We briefly summarize Shor's algorithm for factoring and Simon's algorithm for the hidden XOR-secret problem. A.1 Shor's Algorithm for Factoring Standard number theory reduces factoring N to finding the order of a random element a modulo N , i.e., r > 0

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A. Yao. Quantum Circuit Complexity. In FOCS'93, pp. 351-- 361, 1993. A Shor's Algorithm and Simon's Algorithm We briefly summarize Shor's algorithm for factoring and Simon's algorithm for the hidden XOR-secret problem. A.1 Shor's Algorithm for Factoring Standard number theory reduces factoring N to finding the order of a random element a modulo N , i.e., r ? 0

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