Capabilities and limitations of quantum computers Michele Mosca 1 November 1999 ECC ’99

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)

Overview l A small computer l A quantum computer l Fast quantum algorithms l Limitations l Are they “realistic”?

Computing Model Acyclic circuits of reversible gates

Information and Physics Realisations are getting smaller and faster

A small computer NOT

A small computer

A closer look  NOT

A closer look  NOT

In general

F(x)

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?

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

Efficient algorithms Shor: Find., Generalisations: Find.,

Further generalisation Hidden Subgroup Problem: Find

Another algorithm Hidden Affine Functions: Find using only m evaluations of (instead of n+1) (D,BV,CEMM,H,M)

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’)

Counting Estimate with accuracy (vs. applications classically) Use only applications of. (BBHT,BHT,M,BHMT, ‘amplitude estimation’)

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:

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?

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

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