Quantum Computation: Quantum Computation: A Grand Mathematical A Grand Mathematical Mathematical Mathematical Challenge for the Twenty Challenge for the Twenty- -First Century First Century Why Why Century and the Millennium, Century and the Millennium, Samuel J. Lomonaco, Jr. Samuel J. Lomonaco, Jr. Quantum Quantum (editor), AMS PSAPM/58, (2002). (editor), AMS PSAPM/58, (2002). Algorithms ? Algorithms ? Quantum Computation and Information, Quantum Computation and Information , Samuel J. Samuel J. Lomonaco, Jr. and Howard E. Brandt (editors), (editors), AMS AMS Lomonaco, Jr. and Howard E. Brandt CONM/305, (2002). CONM/305, (2002). Why Quantum Algorithms ? Why Quantum Algorithms ? Why Quantum Algorithms ? Why Quantum Algorithms ? Samuel J. Lomonaco, Jr. Samuel J. Lomonaco, Jr. Dept. of Comp. Dept. of Comp. Sci Sci. & Electrical Engineering . & Electrical Engineering University of Maryland Baltimore County University of Maryland Baltimore County Baltimore, MD 21250 Baltimore, MD 21250 Why Quantum Devices ? Why Quantum Devices ? Email: Lomonaco@UMBC.EDU WebPage: http://www.csee.umbc.edu/~lomonaco • L • L- -O O- -O O- -P P 1
New Quantum Algorithms New Quantum Algorithms A Continuous Variarble A Continuous Variarble Shor Shor Algorithm Algorithm • Wandering • Wandering Shor Shor Algorithms Algorithms This algorithm finds the hidden period of an This algorithm finds the hidden period of an • Continuous Variable • admissible map admissible map Continuous Variable Shor Shor Algorithms Algorithms → • Lifted (Purified) • f : � � Lifted (Purified) Shor Shor • • Quantum Circle Algorithms I double dare Yanhua Yanhua to to Quantum Circle Algorithms I double dare implement this in optics implement this in optics • Dual Lomonaco, Jr., Samuel J., and Louis H. Lomonaco, Jr., Samuel J., and Louis H. • Dual Shor Shor Algorithm Algorithm Kauffman, A Continuous Variable Kauffman, A Continuous Variable Shor Shor Algorithm Algorithm, arXiv:quant , arXiv:quant- -ph/0210141 ph/0210141 • Functional integral Quantum Algorithms • Functional integral Quantum Algorithms (highly speculative) (highly speculative) Distributed Distributed Quantum Quantum Algorithms ? Algorithms ? A Distributed Quantum Algorithms A Distributed Quantum Algorithms Anocha Yimsiriwattana Yimsiriwattana Anocha Distributed Quantum Computing Distributed Quantum Computing Yimsiriwattana, , Anocha Anocha, and Samuel J. , and Samuel J. Yimsiriwattana Lomonaco, Jr., Generalized GHZ States Lomonaco, Jr., Generalized GHZ States Why ?? Why ?? and Distributed Quantum Computing, and Distributed Quantum Computing , arXiv:quant- arXiv:quant -ph/0402148 ph/0402148 • Distributed computing is one path to • Yimsiriwattana, Yimsiriwattana , Anocha Anocha, and Samuel J. , and Samuel J. Distributed computing is one path to Lomonaco, Jr., Distributed quantum Lomonaco, Jr., Distributed quantum scalable quantum computing scalable quantum computing computing: A distributed Shor computing: A distributed Shor, , • Provides a mechanism for isolating the • Provides a mechanism for isolating the algorithm, arXiv:quant- algorithm, arXiv:quant -ph/0403146 ph/0403146 problem of decoherence decoherence problem of 2
Conclusion Overview Overview Conclusion The computational cost of transforming a quantum The computational cost of transforming a quantum algorithm into a distributed quantum algorithm algorithm into a distributed quantum algorithm It really pays to distribute a quantum It really pays to distribute a quantum • Space Complexity Overhead: • algorithm. algorithm. Space Complexity Overhead: The additional space The additional space overhead is insignificant, i.e., the number of overhead is insignificant, i.e., the number of • The problem of • required additional qubits required additional qubits is 5 and independent of is 5 and independent of The problem of decoherence decoherence is isolated is isolated the number of qubits qubits of the non of the non- -distributed distributed the number of to the individual small computing nodes in to the individual small computing nodes in algorithm algorithm the network, where decoherence decoherence is most is most the network, where • • Time Complexity Overhead: easily handled. easily handled. Time Complexity Overhead: There is a resulting There is a resulting a linear slowdown of the non a linear slowdown of the non- -distributed algorithm distributed algorithm • As a result, the need for costly quantum • • But • As a result, the need for costly quantum But … … the rate of slowdown is bounded above by a the rate of slowdown is bounded above by a error correction is reduced by at least one error correction is reduced by at least one constant ( = 9 constant ( = 9 ) ) which is which is independent of the number independent of the number order of magnitude. order of magnitude. of qubits qubits in the non in the non- -distributed. distributed. of Distributed Distributed We Will Show How to Use Quantum We Will Show How to Use Quantum Quantum Quantum Entanglement for Distributed Entanglement for Distributed Computing Computing Control of Quantum Control of Quantum Architecture Architecture Algorithms Algorithms Architecture Architecture Distributed Quantum Computing Distributed Quantum Computing • • By a distributed quantum computer, we Each column represents Each column represents By a distributed quantum computer, we quantum register quantum register a separate computer a separate computer mean mean a network of quantum computers a network of quantum computers interconnected by quantum and classical interconnected by quantum and classical channel channel channels channels qubits qubits • The • The distributed computing paradigm distributed computing paradigm classical classical computer provides an effective way to utilize a provides an effective way to utilize a computer number of small quantum computers number of small quantum computers quantum channels quantum channels classical channels classical channels 3
Section 3 Section 3 Distributed The Application of The Application of Distributed Generalized GHZ States Generalized GHZ States Quantum Quantum and and Cat- Cat -Like States Like States Computing Computing to to Distributed Quantum Computing Distributed Quantum Computing Primitives Primitives Terminology Terminology Key Idea Key Idea A Generalized A Generalized GHZ GHZ state state is a quantum state is a quantum state Use Entanglement to Distribute Control Use Entanglement to Distribute Control of the form of the form ’s s ’s s n 0 0 ’ n 1 1 ’ n n A A generalized GHZ generalized GHZ state can be state can be used to create a used to create a “ “cat cat- -like like” ” state, state, + 00 � 0 11 � 1 which can in turn be used can in turn be used to to distribute distribute which control. . 2 control A Cat A Cat- -Like Like state state is a quantum state of the is a quantum state of the form form α + β 00 � 0 11 � 1 2 Qubit 2 Qubit Entangling Gate Entangling Gate 3 3 Qubit Qubit Entangling Gate Entangling Gate H H = = 0 0 + 00 11 + 000 111 0 2 2 0 0 4
Important Fact Important Fact n n Qubit Qubit Entangling Gate Entangling Gate H Generalized GHZ states are constructed Generalized GHZ states are constructed from EPR pairs by applying only local from EPR pairs by applying only local = operations. operations. • Method 1: • Method 1: Via Via nonlocal nonlocal CNOTS CNOTS • Method 2: • Method 2: Via entanglement swapping Via entanglement swapping So all network quantum So all network quantum channels are EPR channels ! channels are EPR channels ! Non- Non -local CNOT Gate local CNOT Gate Observations Observations • The • α 〉 + β 〉 | 0 |1 The CNOT gate CNOT gate and and the the set of all set of all Z one- one -qubit qubit gates gates are universal are universal | 〉 0 M • If we can implement a non • If we can implement a non- -local CNOT, local CNOT, | 0 〉 X H M then a distributed version then a distributed version of any unitary of any unitary | t 〉 transformation can be implemented transformation can be implemented J. Eisert J. Eisert, K. Jacobs, P. , K. Jacobs, P. Papadopoulus Papadopoulus, and M.B. , and M.B. Plenio Plenio, , “ “Optimal local implementation of non Optimal local implementation of non- -local quantum gates local quantum gates” ”. . Phys. Rev. A, 62, 052317 (2000) Phys. Rev. A, 62, 052317 (2000) Generalized Non- -local CNOT local CNOT Generalized Non Distributing a Control Line via Distributing a Control Line via Cat- -like State like State Cat Cat- Cat -Creator Creator Control Control Disentangler Disentangler α 〉 + β 〉 | 0 | 1 Z Cat Cat- -Like Like | 0 〉 M α 〉 + β 〉 | 000 |111 + C C | 0 〉 X H M H H A A | 0 〉 X H M N N N N | 0 〉 E E X H M L L | t 〉 | t 〉 State to be State to be Cat- Cat -like like α 〉 〉 + β 〉 〉 | 000 | 0 | 111 X | 1 Controlled Controlled α 〉 + β 〉 | 0000 |1111 5
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