P. W. Shor, "Quantum Computing," Documenta Mathematica, vol. Extra Volume ICM, pp. 467486, 1998, http://east.camel.math.ca/ EMIS/ journals / DMJDMV/ xvol-icm/ 00/ Shor.MAN.html.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....characteristics of QEA. Concluding remarks follow in Section 5. 2 Quantum inspired Evolutionary Algorithm (QEA) QEA is based on the concepts of quantum bits and superposition of states [5] The smallest unit of information stored in a two state quantum computer is called a quantum bit or qubit [7, 8, 9]. A qubit may be in the 1 state, in the 0 state, or in any superposition of the two. The classical representation can be broadly classified as: binary, numeric, and symbolic [10] QEA uses a new representation that is based on the concept of qubits. QEA with the qubit representation has a ....

P. W. Shor, "Quantum Computing," Documenta Mathematica, vol. Extra Volume ICM, pp. 467486, 1998, http://east.camel.math.ca/ EMIS/ journals / DMJDMV/ xvol-icm/ 00/ Shor.MAN.html.

....the late 1980s [8] Many efforts on quantum computers have progressed actively since the early 1990s because these computers were shown to be more powerful than classical computers on various specialized problems. There are well known quantum algorithms such as Shor s quantum factoring algorithm [9], 10] and Grover s database search algorithm [11] 12] Research on merging evolutionary computing and quantum computing has been started since late 1990s. They can be classified into two fields. One concentrates on generating new quantum algorithms using automatic programming techniques such as ....

P. W. Shor, "Quantum computing," Doc. Mathematica, vol. Extra Volume ICM, pp. 467--486, [Online]. Available: http://east.camel.math.ca/ EMIS/ journals/ DMJDMV/ xvol-icm/ 00/ Shor.MAN.html, 1998.

....that could solve computational problems faster than classical algorithms, to date, only a few quantum algorithms are known. Nevertheless, quantum computing is attracting serious attention, since its superiority was demonstrated by a few quantum algorithms such as Shor s quantum factoring algorithm [1, 2] and Grover s database search algorithm [3, 4] Shor s algorithm finds the prime factors of an n digit number in polynomial time, while the best known classical factoring algorithms require time O ###### . Grover s database search algorithm can find an item in an unsorted list of n items ....

P. W. Shor, "Quantum Computing," Documenta Mathematica, vol. Extra Volume ICM, pp. 467486, 1998, http://east.camel.math.ca/ EMIS/ journals / DMJDMV/ xvol-icm/ 00/ Shor.MAN.html.

....theory the smallest conveying information unit is the bit. The counterpart unit in quantum information is called the quantum bit the qubit. Its state can be described by means of the state # = # 0# # 1#, where #, # C refers to the complex probability amplitudes and # = 1 [7, 8, 6]. The expression denotes the probability that after measuring the qubit it can be found in computational base, and shows the probability to be in computational base 1#. In more general description an N bit qregister is set up from qbits spanned by i = 0 . M 1) ....

....To the irreversiblity of transformation, U must be unity U 1 = U . The quantum registers can be set in a general state using quantum gates which can be represented by means of a unitary operation, described by a quadratic matrix. Applying four basic gates any states can be prepared [7]. 4 System Model In DS CDMA systems an information bearing bit is encoded by means of a user specific code with length of the processing gain (PG) 1] In case of uplink communication we assume perfect power control. In the receiver side it is not required synchronization between input signals ....

P.W. Shor, "Quantum computing," Documenta Mathematica, vol. 1-1000, 1998. Extra Volume ICM.

....information theory the smallest information conveying unit is the bit. The counterpart unit in quantum information is called the quantum bit , the qubit. Its state can be described by means of the state . X ) refers to the complex probability amplitudes and G G 5 [8], 9] 6] The expression G denotes the probability that after measuring the qubit it can be found in computational base , and G shows the probability to be in computational base . In more general description an bit quantum register (qregister) is set up from qubits ....

P.W. Shor, "Quantum computing," Documenta Mathematica, vol. 1-1000, 1998. Extra Volume ICM.

....information theory the smallest conveying information unit is the bit. The counterpart unit in quantum information is called the quantum bit the qubit. Its state can be described by means of the state # = # 0# # 1#, where #, # C refers to the complex probability amplitudes and = 1 [7, 8, 6]. The expression denotes the probability that after measuring the qubit it can be found in computational base, and shows the probability to be in computational base 1#. In more general description an N bit qregister is set up from qbits Say ket #. Ferenc Balzs: Non Coherent MUD ....

....To the irreversiblity of transformation, U must be unity U 1 = U . The quantum registers can be set in a general state using quantum gates which can be represented by means of a unitary operation, described by a quadratic matrix. Applying four basic gates any states can be prepared [7]. 4. System Model In DS CDMA systems an information bearing bit is encoded by means of a user specific code with length of the processing gain (PG) 1] In case of uplink communication we assume perfect power control. In the receiver side it is not required synchronization between input signals ....

P.W. Shor, "Quantum computing," Documenta Mathematica, vol. 11000, 1998. Extra Volume ICM.

.... and the quantum mechanical properties of these systems can be harnessed to compute certain functions more efficiently than is possible on any classical computer [1] For example, Shor s quantum factoring algorithm finds the prime factors of an n digit number in time O(n 2 log(n) log log(n) [2], while the best known classical factoring algorithms require time O(2 n 1 3 log(n) 2 3 ) and many researchers doubt the existence of polynomial time classical factoring algorithms [3, 4] An unambiguous case of the superiority of quantum computation was provided by Grover, who showed how a ....

.... does not beat the best classical To appear in Proceedings of the 1999 Congress on Evolutionary Computation 6 U THETA [0] 5pi 11) HADAMARD [1] U2 [0] 4pi 3pi 14 pi 14 12pi 13) ORACLE [0 (in high) 1 (in low) 2 (out) HADAMARD [1] MEASURE 1 [1] HADAMARD [0] U2 [0] 0 0 pi 5 2pi) HADAMARD [2] U THETA [2] 11pi 9) U THETA [0] 3pi 4) U THETA [0] 4pi 15) MEASURE 0 [0] MEASURE 0 [2] MEASURE 1 [2] Figure 2: New evolved algorithm for AND OR Function Error Prob. Function Error Prob. 0 0 0 0 0.0075. 1 0 0 0 0.2922. 0 0 0 1 0.2751. 1 0 0 1 0.2936. 0 0 1 0 0.2751. 1 0 1 0 ....

[Article contains additional citation context not shown here]

Peter W. Shor, "Quantum computing," Documenta Mathematica, vol. Extra Volume ICM, pp. 467--486, 1998, http://east.camel.math.ca/EMIS/ journals /DMJDMV/xvo l-icm/00/ Shor.MAN.ps.gz.

....if built, will be able to compute certain functions more efficiently than is possible on any classical computer. For example, Shor s quantum factoring algorithm finds the prime factors of a number in polynomial time, while the best known classical factoring algorithms require exponential time [11]. Another important example was provided by Grover, who showed how a quantum computer could find an item in an unsorted list of n items in O( # n) steps, while classical algorithms require O(n) steps [12] A brief introduction to the core ideas of quantum computing was provided in an earlier ....

P. W. Shor, "Quantum computing," Documenta Mathematica, vol. Extra Volume ICM, pp. 467--486, 1998. 6

....to the tensor product of n copies of the 2 dimensional Hilbert space in which a single qubit lives. If v = v 1 ; v 2 ; v n ) where v i 2 Z 2 = f0; 1g for i = 1; 2; n, then e v = e v 1 Omega e v 2 Omega Delta Delta Delta Omega e vn : Why the tensor product Peter Shor [17] says: One of the fundamental principles of quantum mechanics is that the joint quantum state space of two systems is the tensor product of their individual state spaces. The tensor product of two spaces is the universal bilinear product of the spaces. If fe 1 ; e mg and ff 1 ; f ....

....m 1 ; b n Gammak i z V . We will follow Cleve and Gottesman [6] to build a quantum circuit to accomplish this encoding. As Shor says, a quantum circuit is built out of logical quantum wires, each corresponding to one of the n qubits, and quantum gates, each acting on one or two wires [17]. A quantum gate is a unitary transformation, since these correspond to the possible physical transformations of a quantum system. Shor assumes that each gate acts on either one or two wires; that is, each maps 1 qubit to 1 qubit or 2 qubits to 2 and acts as the identity on the remaining qubits. ....

[Article contains additional citation context not shown here]

P. W. Shor, Quantum Computing, ICM 1998 lecture, http://www.research.att.com/ ~ shor/papers/ICM.ps.

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Shor, Peter W. Quantum computing. Proceedings of the International Congress of Mathematicians, Vol. I (Berlin,

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