Quantum Computation Dr Austin Fowler Centre for Quantum Computer Technology New Scientist, 10/11/07
Overview what is a quantum computer? bits vs qubits superpositions and measurement implementations why quantum compute? existing computers are great, however... quantum parallelism and factoring other promising quantum algorithms quantum cryptography primary challenge: quantum errors correction techniques open problems conclusion: we need you! dilution refrigerator capable of cooling samples to a hundredth of a degree above absolute zero
What is a quantum computer? first: what is a classical computer? NAND A B Q 0 0 1 0 1 1 1 0 1 1 1 0 definition 1: device capable of executing NAND NAND is universal, right?
What is a classical computer? much more in the figure initialisation fanout internal memory transport readout NAND full computer also has clock external memory program controller
What is a quantum computer? introducing CNOT introducing U A B C D 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0
What is a quantum computer? above circuit has special name: Toffoli implies quantum computing contains classical computing
What is a quantum computer? superpositions: measurements no classical equivalents
What is a quantum computer? implementations: ion traps qubit: electron excited/not CNOT: collide atoms U: lasers initialisation/readout: lasers + sensitive optical detectors fanout: impossible transport: electromagnetic shuttling internal memory: 10-30 seconds clock, external memory, program controller: all classical
What is a quantum computer? flux qubits liquid NMR optical lattices phosphorus in silicon
What is a quantum computer? linear optics on-chip optics quantum dots diamond
What is a quantum computer? ultimately, implementation is someone else s problem re-introducing CNOT re-introducing U A B C D 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0 also: initialisation, readout, memory, transport everything hard to implement, at least one error rate 10%+
Why quantum compute? classical computers are pretty awesome! stats of the Cray Jaguar (world s most powerful computer): 1.75 petaflops 10 petabytes of storage 180,000 processing cores 578 TB/s memory bandwidth 360 terabytes of RAM 284 GB/s I/O bandwidth surely that s enough for anything?
Why quantum compute? not enough for factoring a 2048 bit number! N N 1 N 2 inefficient on a classical computer transform to f(x) = m x mod N = f(x+x p ) x p fire up quantum computer! calculate measure Quantum Fourier Transform (QFT) x p N N 1 N 2 efficient on a quantum computer!
Why quantum compute? full dynamics of materials every atom contains many quantum d.o.f. need at least 2 d.o.f. space to represent quantum properties even 10 petabytes is at most only 60 d.o.f. meaningful simulations of HTSC impossible quantum chemistry, materials design, particle physics
Why quantum compute? code breaking particle physics quantum key distribution random walks materials simulation BQP = NP? early universe quantum chemistry knot theory group theory
Primary challenge: quantum errors initialisation, memory (I), unitary (U), CNOT, measurement at least one 10%+ error rate operation in existing implementations intolerable... using existing quantum error correction (QEC) techniques
Quantum error correction what is a quantum error? hard to correct! much easier if errors discrete bit-flip phase-flip Peter Shor noted: arbitrary error just superposition of X and Z! need to define stabilisers to proceed
Stabilisers
Surface code most powerful QEC code invented to date requires 2D lattice of qubits only requires nearest neighbor interactions circuits capable of measuring the sign of a stabiliser
Quantum error correction
Quantum error correction Record time and position of changed syndromes Match closest pairs Apply corrective operations to spacelike edges Works very well, threshold error rate ~1%
The story so far... a quantum computer will be a 2D lattice of qubits optical lattices superconducters ion traps many others current experimental error rates 10%+ robust computation via the surface code current maximum tolerable error rate ~1%
Open problems make the classical processing much faster O(n 3 ) parallelise and localise the classical processing raise the maximum tolerable error rate! lower overhead
Faster classical processing minimum weight perfect matching algorithm O(n 3 ) time on n nodes basic idea: chose unmatched node O(n) check each edge of tree to see if tight O(n) grow tree using tight edges till another unmatched node found O(n) augment
Faster classical processing (nearly) complete weighted graph takes O(n 2 ) time just to input unnecessary! Viewed from any node, every nearest neighbour casts a shadow only need constant number of edges to obtain optimal solution
Faster classical processing given tree consisting of n t nodes each with m edges on average, need O(n t m) time to calculate tightest edge unnecessary! Store edges of each node in Fibonacci heaps prioritised by tightness can merge n h heaps in O(n h ) time can find tightest edge in O(1) time
Faster classical processing as more nodes are matched, remaining nodes become isolated requiring larger trees unnecessary! Match nodes systematically in space and time and grow trees preferentially in the direction of unmatched nodes none of these ideas have been implemented despite O(n 3 ) O(n) advantage
Parallelise and localise basic idea: not implemented
Tolerate more error currently, X/Z errors corrected independently an error means one of X, Z, XZ if we know X error present, 50% chance Z error present not implemented
Lower overhead details beyond the scope of this talk highly nontrivial to pack circuits open problems discussed here are just the tip of the iceberg
Summary quantum computers can solve more problems efficiently than classical computers quantum computers are technologically exotic and error prone controlling quantum errors could be done much better with help from computer science quantum computing needs you! Interested? Austin Fowler: afowler@unimelb.edu.au Adrian Pearce: adrianrp@unimelb.edu.au