Capabilities and limitations of quantum computers

Capabilities and limitations of quantum computers Michele Mosca 1 November 1999 ECC ’99 mmosca@cacr.math.uwaterloo.ca

What I’m not talking about • Quantum Communication Theory (reduce the complexity of distributed computation tasks; ask Alain Tapp) • Quantum Information Security (quantum key exchange; security based on uncertainty principle and not computational assumptions)

Overview • A small computer • A quantum computer • Fast quantum algorithms • Limitations • 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 NOT

In general

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) Find using 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 . , Find . Generalisations: Find . , 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 Use only applications of . (BBHT,BHT,M,BHMT, ‘amplitude estimation’) (vs. applications classically)

Limitations No luck with: • Square root speed up for serial algorithms • Graph automorphism/isomorphism • Short vectors in a lattice • NP-complete problems (e.g. minimum codeword, graph colouring, subset sum, …)

What about implementations? • 1-7 qubits using NMR technology • 1-2 qubits using ion traps • 1-2 qubits using various other quantum technologies • Scaling is very hard! • Is the problem technical or fundamental?

Technical or Fundamental? • Noise, “decoherence”, imprecision are detrimental • Similar problems exist in “classical” systems • Theory of linear error correction and fault tolerant computing can be generalised to the quantum setting (Shor, Steane, etc.) • Using “reasonable” physical models, there exist fault-tolerant schemes for scalable quantum computing

Summary • Quantum Computers are a natural generalisation of “classical” computers • Quantum algorithms: Factoring, Discrete log, Hidden Subgroup, Hidden Affine Functions, Searching, Counting • Small implementations exist • Scaling is difficult, but seems to be a technological (not fundamental) problem

Capabilities and limitations of quantum computers