When Physics and Computer Science Collide

1. When Physics and Computer Science Collide Dave Bacon Department of Computer Science & Engineering University of Washington, Seattle, WA

2. A Critical Ghost All papers on quantum computing should carry a footnote: “This proposal, like all proposals for quantum computation, relies on speculative technology, does not in its current form take into account all possible sources of noise, unreliability and manufacturing error, and probably will not work.” Rolf Landauer IBM • Maintenance of giganto-coherence? • Faulty quantum gates? • Do we understand the physics of quantum errors in the system? Nature abhors a quantum computer?

3. Quantum Error Correction! Rolf Landauer IBM Threshold Theorem for Fault-Tolerant Quantum Computation

4. Noisy Cell Phone Hello? Hello? Hello? Hello? I have a flat tire. I said, I have a flat tire! A flat tire. No, I’m not trying to flatter you..No, you’re not getting fatter. I have a flat tire! Communication over a noisy CHANNEL can be overcome via ENCODING “Hello?” = “Hello? Hello? Hello? Hello?” [using redundancy to encode “Hello”]

5. Simple Repetition Code Encode: 0 0 n copies 1 1 Binary Symmetric Channel No encoding: b b b b b measure Encoding (n=3): decode and correct encode measure Probability of error

6. 1994 Reasons to be a Pessimist No cloning: Quantum Cloning Machine “A single quantum cannot be cloned,” Wootters and Zurek, Nature, 1982 No quantum repetition code: Measurement destroys coherence: How can one decode without destroying the information?

7. Unrealistic Realistic Channel

8. Baby Steps WWCCD? (What Would Classical Coders Do?) measure b b b 0 000, 1  111 0 error #@% 0 b encode error decode fix identities 100  111101  110 110 = b b 1 0 = 1 0 b 1 1

9. Naïve Lets be naïve, take classical and move over to quantum error encode decode fix ? error fix decode 3. syndrome • encoded into subspace: 2. errors take to orthogonal subspaces + maintain orthogonality (no-cloning evaded!) 4. operator identities still hold

10. OK Wise Guy What about “phase” errors? …sort of not classical error phase error: Wise guy says “basis change please”: looks like bit flit error in this new basis! H H H H H H phase errors bit flip errors

11. Molly: “I love you, I really love you” Sam: “Ditto.” 3. syndrome • encoded into subspace: 2. errors take to orthogonal subspaces + maintain orthogonality (no-cloning evaded!) encode error decode fix ? H H H H H H error fix decode

12. Encoding Away Your Ills 3 qubit bit flip code 3 qubit phase flip code phase errors act as on bit flip code qubits: define: Shor Code: (Peter Shor, 1995)

13. Inside Shor H H H H H H bit flip code phase flip code

14. system environment Interaction causing DECOHERENCE System evolution environment evolution What is the Problem? Real quantum systems are open quantum systems! Full Hamiltonian: System evolution (by itself) is no longer unitary Coherent properties of system’s quantum information lost

15. Quantum Error Correction error diagnose fix encoded quantum information encoded quantum information “cold” ancilla qubits “hot” ancilla qubits Quantum error correction is essentially (sophisticated) cooling

16. Perfection Through Concatenation failure probability p cp2 c(cp2)2 At kth level of concatenation, probability of an error which destroys the encoded quantum information is and uses qubits

17. The Quantum Hard Drive? Kitaev Do there exist (or can we engineer) quantum systems whose physics guarantees fault-tolerant quantum computation?

18. Kitaev’s Model plaque operators: qubits on links vertex operators: vertex and plaque operators all commute with each other

19. Kitaev’s Model, Continued Hamiltonian plaque operators: vertex operators: ground state has all

20. Topology of Operators plaque operators: products of vertex operators are closed loops of Z’s which are homologically trivial on dual lattice, vertex operators play same rule

21. Degeneracy Hamiltonian plaque operators: vertex operators: qubits plaque operators values vertex operators values every energy level is degenerate

22. Kitaev’s Model plaque operators: vertex operators: anti-commutes with two plaque operators excitation is above ground state  

23. Excitations excitations particles come in pairs (particle/antiparticle) at end of “error” chains two types of particles, X-type (live on vertices of dual lattice) Z-type (live on vertices of the lattice)

24. A Qubit Moving an X or Z type particle on a homologically nontrivial path and annihilating with it’s partner preserves the ground states Action on ground state must be representation of non-Abelian group: Acts as encoded single qubit Pauli operator.

25. Two Qubits Moving an X or Z type particle on a homologically nontrivial path and annihilating with it’s partner preserves the ground states Action on ground state must be representation of non-Abelian group: Acts as encoded single qubit Pauli operator.

26. Anyons Anyone? Phase:

27. Topological Quantum Computing Encode two qubits into the ground state gap If we keep then quantum information encoded into ground state will be “topologically” robust to perturbation. We can perform gates (in this example X and Z) by creating anyon pairs and moving them around the torus. Are these assumptions reasonable?

28. Topological Error Correction Different mode of using Kitaev’s Model, as a component in a fault-tolerant quantum computer, encode into ground state 1. Errors occur 2. Measure vertex and plaque operators 3. Apply recovery operator 4. If created loop is homologically trivial, then we have restored information encoded into ground state. Full analysis: measurements uncertain, recovery may fail, etc. Threshold for fault-tolerant storage of quantum information: [Dennis, Kitaev, Landahl, Preskill 2002] [Wang, Harrington, Preskill 2003]

29. Fault-Tolerant? Encode two qubits into the ground state gap If we keep then quantum information encoded into ground state will be “topologically” robust to perturbation. We can perform gates (in this example X and Z) by creating anyon pairs and moving them around the torus. Are these assumptions reasonable?

30. What is a Computer? Practical question (as opposed to philosophical) What is the phase of matter corresponding to the computer? There are distinct physical reasons why robust classical computation is possible. Hard Drive Integrated Circuit

31. “Dave, may I be excused? My brain is full.”