Topological Quantum Computation

Transcription

1 Texas A&M University October 2010

2 Outline 1 Gates, Circuits and Universality Examples and Efficiency 2 A Universal 3

3 The State Space Gates, Circuits and Universality Examples and Efficiency Fix d Z Definition Let V = C d. The n-qudit state space is the n-fold tensor product: H qc (n) = V V V. v 1 v 2 v n H qc (n) is a n qudit register. Notice: dim H qc (n) = d n.

4 Gates and Circuits Gates, Circuits and Universality Examples and Efficiency A quantum gate is a unitary operator U i U(H qc (n i )) A gate set S = {U i } is a collection of gates. Fix U U(H qc (n)). Definition A quantum circuit for U on S is: G 1,..., G m U(H qc (n)) where G i = I a V U j I b V, U j S, a + b + n i = n and G 1 G 2 G m = U

5 Remarks Gates, Circuits and Universality Examples and Efficiency Remarks Realistically: Thus U i U(H qc (n i )) and U U(H qc (n)) with n i n so G i are local. Sad Fact S is finite so for fixed n only finitely many G i possible. No finite gate set can exactly implement every U U(H qc (n)).

6 Universality Gates, Circuits and Universality Examples and Efficiency Let U U(H qc (n)) and S a gate set. Definition S approximately simulates U within ɛ if there exists G 1,, G m such that G 1 G m U < ɛ Here is the operator norm. Definition If S approximately simulates any U U(H qc (n)) within ɛ for any ɛ > 0 S is universal. In other words, if the set of promotions of S to U(H qc (n)) is dense.

7 Universal Examples Gates, Circuits and Universality Examples and Efficiency The following ( set ) is universal ( for d= 2: {H, ) σ z ±1/4, CNOT } H = , σ z ±1/4 1 0 = e ±πi/ CNOT = Also any entangling U U(H qc (2)) with all of U(H qc (1)) is univeral (Brylinskis Theorem). (A tall order!)

8 Efficiency? Gates, Circuits and Universality Examples and Efficiency Let S be universal, U U(H qc (n)) and ɛ > 0 so that G 1 G m U < ɛ. How big is m? Theorem (Solovay-Kitaev) If S is closed under U i U 1 i then m = O(log 2 1/ɛ). BQP Class of problems efficiently solved by QC: B(ounded error)q(uantum resource)p(olynomial time)

9 Origins A Universal Some anyons Top. Quant. Field Theory Kitaev 97 Freedman 97 Quantum Computer

10 Topological Phases of Matter A Universal Definition (Nayak, et al) a system is in a topological phase if its low-energy effective field theory is a topological quantum field theory. Remarks Anyons are examples of topological phases Hence Freedman-Kitaev model: TQFTs model systems of anyons. a topological quantum computer would be realized on such systems.

11 Topological Quantum Field Theory? A Universal Definition A (unitary) 3D TQFT assigns to any (compact oriented labelled extended) surface (M, l) a (finite-dimensional) Hilbert space: H top (M, l), subject to (many) compatibility axioms. Key: gluing and disjoint sum axioms. Labels L a finite set, 0 L distinguished, with involution x ˆx. Remarks Each component of boundary M is labelled H top (D 2, 0) = C. H top (n) := H top (D 2 \ {z i } n i=1, (0, t,..., t))

12 Two axioms A Universal Axiom (Disjoint Sum) H top ((M 1, l 1 ) (M 2, l 2 )) = H top (M 1, l 1 ) H top (M 2, l 2 ) Axiom (Gluing) If M g is obtained from gluing two boundary circles of M together then H top (M g, l) = x L H top (M, (l, x, ˆx))

13 Example: FQH Liquid Cartoon A Universal Fractional Quantum Hall Liquid electrons/cm 2 H top =H 9 mk defects=quasi-particles 10 Tesla

14 (non-adaptive) A Universal Computation Physics output measure apply gates particle exchange initialize create particles vacuum

15 The Braid Group A Universal A key role is played by: : Definition The braid group B n has generators σ i, i = 1,..., n 1 satisfying: (R1) σ i σ i+1 σ i = σ i+1 σ i σ i+1 (R2) σ i σ j = σ j σ i if i j > 1

16 A Universal In Pictures: Source of Fault-Tolerence 1 i i+1 n i -1 = Multiplication : =

17 Remarks on A Universal Remarks H top (n) is n-particle state space Gate set S is particle exchanges Mathematically, S = {ϕ n (σ i )} where ϕ n : B n U(H top (n)) (ϕ n, H top (n)) are unitary B n -representations. i i+1 i

18 Fibonacci Theory A Universal Two particle types: X 0 = 1 (vacuum) and X 1. X 1 X 1 = 1 X 1 : two fusion channels (non-abelian!). V i k := H top(d 2 \ {z i } k i=1, (i, 1,..., 1)) D2 labeled by i V3 1 = C 2 (qubits!) { More generally, dim Vk i = Fib(n 2) i = 0 Fib(n 1) i = 1 1, 1, 2, 3, 5, 8,..., τ n ( τ) n 5,...

19 Why Universal? A Universal Particle exchange induces the: Definition Jones representation (at 5th roots of unity): ρ 5 n : B n U(Vn 0 ) U(Vn 1 ) U(Vn 0 Vn 1 ) Theorem (Freedman,Larsen,Wang) {ρ 5 n(σ i )} n 1 i=1 is dense in SU(V 0 n ) SU(V 1 n ). Universality follows from Kitaev-Solovay (after some comparisons...)

20 What do TQCs like to compute? A Universal Answer (Approximations to) Link invariants! (at roots of unity). Fibonacci theory: Jones polynomial J L (q) at q = e 2πi/5. L Prob( ) V_L(q)

21 A Universal Complexity of Jones Polynomial Evaluations Classically we have: Theorem (Vertigan,Wocjan-Yard) Exact computation of J L (q) at q = e 2πi/r is: { FP r = 3, 4, 6 FP #P else And on a quantum computer: Theorem Approximation of J L (q) at q = e 2πi/r is BQP-complete for r 3, 4, 6.

22 Head-to-head State space a tensor product: dim H qc (n) = d n Gates are local: G i acts on n i n qu-dits in H qc (n) Problem: decoherence many algorithms State space not a tensor product: dim H top (n) c n Gates are global: ϕ n (σ i ) smeared across H top (n) Fault-tolerant few algorithms

23 Equivalence Theorem (Freedman,Kitaev,Wang) (Universal) QCM simulates TQCs efficiently. Theorem (Freedman,Larsen,Wang) (at least one) TQC simulates QCM efficiently. Main Issue H qc (n) = V n while H top (k) = i W n i

24 Mathematics of Equivalences Question Given U f U(H qc (n)) how to (approximately) simulate U f on H top (k)? Given U β U(H top (k)) how to simulate U β on H qc (n)? Answer Efficiently embed V n into H top (k(n)). Efficiently embed H top (k) into V n(k). Each uses gluing axiom Inconvenient Truth Leads to leakage errors. (next time...)

25 Best of Both Worlds? Question Is there a model that is: Universal purely topological (fault-tolerant) and (explicitly) local? Why? QC algorithms in fault-tolerant universal setting!

26 Thank You!