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Capabilities and limitations of quantum computers Michele Mosca mmosca@cacr.math.uwaterloo.ca 1 November 1999 ECC ’99
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What I’m not talking about l Quantum Communication Theory (reduce the complexity of distributed computation tasks; ask Alain Tapp) l Quantum Information Security (quantum key exchange; security based on uncertainty principle and not computational assumptions)
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Overview l A small computer l A quantum computer l Fast quantum algorithms l Limitations l Are they “realistic”?
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Computing Model Acyclic circuits of reversible gates
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Information and Physics Realisations are getting smaller and faster
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A small computer NOT
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A small computer
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A closer look NOT
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A closer look NOT
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In general
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F(x)
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Quantum computers Note that it becomes exponentially difficult (classically) to keep track of an n-qubit system after t operations, but to implement quantumly only requires n qubits and t steps! (Feynman ’82, Deutsch ’85) Can we exploit this apparent computational advantage?
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Efficient algorithms (Deutsch ’85) Findusing only 1 evaluation of (Deutsch, CEMM, Tapp; implemented in NMR by Jones&M, Chuang et al.) Bernstein&Vazirani, Simon came up with relativized separations between P and QP
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Efficient algorithms Shor: Find., Generalisations: Find.,
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Further generalisation Hidden Subgroup Problem: Find
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Another algorithm Hidden Affine Functions: Find using only m evaluations of (instead of n+1) (D,BV,CEMM,H,M)
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Searching and Counting Find Suppose algorithm succeeds with probability (e.g. ). We can iterate and times to find such an. i.e. SQUARE ROOT speed-p (Grover, BBHT,BH, ’amplitude amplification’)
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Counting Estimate with accuracy (vs. applications classically) Use only applications of. (BBHT,BHT,M,BHMT, ‘amplitude estimation’)
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Limitations l Square root speed up for serial algorithms l Graph automorphism/isomorphism l Short vectors in a lattice l NP-complete problems (e.g. minimum codeword, graph colouring, subset sum, …) No luck with:
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What about implementations? l 1-7 qubits using NMR technology l 1-2 qubits using ion traps l 1-2 qubits using various other quantum technologies l Scaling is very hard! l Is the problem technical or fundamental?
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Technical or Fundamental? l Noise, “decoherence”, imprecision are detrimental l Similar problems exist in “classical” systems l Theory of linear error correction and fault tolerant computing can be generalised to the quantum setting (Shor, Steane, etc.) l Using “reasonable” physical models, there exist fault-tolerant schemes for scalable quantum computing
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Summary l Quantum Computers are a natural generalisation of “classical” computers l Quantum algorithms: Factoring, Discrete log, Hidden Subgroup, Hidden Affine Functions, Searching, Counting l Small implementations exist l Scaling is difficult, but seems to be a technological (not fundamental) problem
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