Minor-Embedding in Adiabatic Quantum Optimization

Transcription

1 Minor-Embedding in Adiabatic Quantum Optimization Vicky Choi Department of Computer Science Virginia Tech Nov, 009

2 Outline Adiabatic Quantum Algorithm -SAT QUBO Minor-embedding Parameter Setting Problem Adiabatic Quantum Architecture Design

3 Outline Adiabatic Quantum Algorithm -SAT QUBO Minor-embedding Parameter Setting Problem Adiabatic Quantum Architecture Design

4 System Hamiltonian: Adiabatic Quantum Algorithm H(t) = ( s(t))h init + s(t)h final for t [0, T ], s(0) = 0, s(t ) =. Initial Hamiltonian: H(0) = H init ground state known (easy to construct) Final Hamiltonian: H(T ) = H final ground state encodes the answer to the desired optimization problem Evolution path: s : [0, T ] [0, ], e.g., s(t) = t T T : running time of the algorithm Adiabatic Theorem: If H(t) evolves slowly enough (or T is large enough), the system remains close to the ground state of H(t) SOLUTION: ground state of H(T ) = H final!

5 Example (Final Hamiltonian for -SAT) For clause C = x i x j x k, define h C = ( + σz i )( σ z j )( + σ z k) h C x = { 0 if x satisfies C H final = otherwise C Clauses Eigenvalues and Eigenvectors of σ z h C Recall: { σ z 0 = 0 σ z = (I σ z ) 0 = 0 0, (I σ z ) = (I + σ z ) = 0, (I + σ z ) 0 = 0

6 Example (Initial Hamiltonian for -SAT) n H inital = i σi x i= where i is the number of clauses in which x i appears. Eigenvalues and Eigenvectors of σ x Recall: σ x = [ ] 0 0 eigenvalue eigenvector 0 + 0

7 Adiabatic Theorem: Running time vs Spectral gap For s(t) = t T. If T is large enough: ( ) poly(n) T = O gap, min then the system remains close to the ground state of H(t).

8 Adiabatic Theorem: Running time vs Spectral gap For s(t) = t T. If T is large enough: ( ) poly(n) T = O gap, min then the system remains close to the ground state of H(t). NOTE: /gap min is polynomial (exponential resp.), T is polynomial (exponential resp.). Question: What s gap min for the -SAT problem?

9 Adiabatic Time Complexity Computing spectral gaps directly is as hard as solving original problem Numerical methods: integration of Shrödinger equation, exact diagonization (spectral gaps), quantum monte carlo (spectral gaps) Ground state decomposition (GSD): unveil the quantum evolution blackbox (another talk )

10 Solving MIS On an Adiabatic Quantum Computer Recall: mis(g) = max Y(x,..., x n ) Max Y(x,..., x n ) = x i i V(G) Min E(s,..., s n ) = i V(G) h i s i + ij E(G) ij E(G) J ij x i x j J ij s i s j where h i = j nbr(i) J ij, for i V(G). In general, Quadratic Unconstrained Binary Optimization (QUBO) problem: Max Y(x,..., x n ) = i V(G) c i x i ij E(G) J ij x i x j

11 Solving MIS On an Adiabatic Quantum Computer Recall: mis(g) = max Y(x,..., x n ) Max Y(x,..., x n ) = x i i V(G) Min E(s,..., s n ) = i V(G) h i s i + ij E(G) ij E(G) J ij x i x j J ij s i s j where h i = j nbr(i) J ij, for i V(G). In general, Quadratic Unconstrained Binary Optimization (QUBO) problem: Max Y(x,..., x n ) = i V(G) c i x i ij E(G) J ij x i x j

12 Final Hamiltonian H final for Ising Problem Ising Problem: Min E(s,..., s n ) = h i s i + J ij s i s j i V(G) ij E(G) G : input graph H final = h i σi z + J ij σi z σj z. G : hardware graph i V(G) ij E(G)

13 Input Graph (G) vs Hardware Graph (U) Input Graph G Hardware Graph (U) What if G is NOT a subgraph of the hardware graph U?

14 Outline Adiabatic Quantum Algorithm -SAT QUBO Minor-embedding Parameter Setting Problem Adiabatic Quantum Architecture Design

15 IDEA: a b logical qubit: vertex in the input graph physical qubit: vertex in the hardware graph Question: Can two physical qubits a, b ACT AS one logical qubit? If so, how?

16 Definition The minor-embedding of G is defined by φ : G U such that each vertex i in V(G) is mapped to a (connected) subtree T i of U; there exists a map τ : V(G) V(G) E(U) such that for each ij E(G), there are corresponding i τ(i,j) T i and j τ(j,i) T j with i τ(i,j) j τ(j,i) E(U). G φ(g) = G emb

17 Definition The minor-embedding of G is defined by φ : G U such that each vertex i in V(G) is mapped to a (connected) subtree T i of U; there exists a map τ : V(G) V(G) E(U) such that for each ij E(G), there are corresponding i τ(i,j) T i and j τ(j,i) T j with i τ(i,j) j τ(j,i) E(U). G φ(g) = G emb

18 Minor-Embedding Reduction G emb : a minor-embedding of G Energy of the embedded Ising Hamiltonian: E emb (s,..., s N ) = h is i + J ijs i s j where V(G emb ) = N Reduction Minimum of E emb Minimum of E i V(G emb ) ij E(G emb )

19 Why does the minor-embedding work? Ferromagnetic(FM)-coupling: F < 0 E emb (s,..., s N ) = i V(G) i k V(T i ) h i k s ik + F pq i i pi q E(T i ) ij E(G) s ip s iq + J ij s iτ(i,j) s jτ(j,i) where i k V(T i ) h i k = h i Question: What should h s and F s be?

20 An Easy Upper Bound F i < ( h i + Idea: penalty or multiplier method Change the constrained problem: min E constrained (s,..., s N ) = subject to j nbr(i) i V(G) i k V(T i ) J ij ) h i k s ik + ij E(G) s ip s iq = for all i p i q E(T i ), i V(G) J ij s iτ(i,j) s jτ(j,i) Meaning: Spins of physical qubits that correspond to the same logical qubit are of the same sign.

21 Theorem (A Tighter Bound) (For chains topological-minor-embedding) ( def For i V(G), C i = j nbr(i) J ij h i ). Set { h i j k = sign(h i ) τ(j,i) Onbr(i k ) J ij C i / j τ(j,i) Onbr(i k ) J ij i k is a leaf of T i otherwise and F e i < C i /, for each e E(T i ). Then there is one-one correspondence between arg min E and arg min E emb. V. Choi. Minor-embedding in adiabatic quantum computation: I. The parameter setting problem. Quantum Inf. Processing.,, 9 09, 00.

22 QUBO Ising Model J i J i c i J i h i = ( J ij c i ) J i J id J id embed (J i + J i ) c i Ji 0 F F J i J i J i (J i(d ) + J id ) ci (J>0) F < c i

23 Outline Adiabatic Quantum Algorithm -SAT QUBO Minor-embedding Parameter Setting Problem Adiabatic Quantum Architecture Design

24 Physical Qubit & Coupler (D-Wave Systems Inc.) Known physical constraints proximity Coupler (=Edge) length can not be too long Each qubit (=vertex) is coupled (adjacent) to only a small constant of other qubits The wire of a qubit can be stretched. The shape of each qubit does not need to be a small circle.

25 Hardware graph Geometric graph (or layout) Edge length is bounded (all adjacent qubits within a bounded distance) Bounded degree Non-planar: crossing is allowed Definition Given a family F of graphs, a (host) graph U is called F-minor-univeral if for any graph G F, there exists a minor-embedding of G in U. Complete graph K n Sparse graphs Related work

26 Hardware graph Geometric graph (or layout) Edge length is bounded (all adjacent qubits within a bounded distance) Bounded degree Non-planar: crossing is allowed Definition Given a family F of graphs, a (host) graph U is called F-minor-univeral if for any graph G F, there exists a minor-embedding of G in U. Complete graph K n Sparse graphs Related work

27 Hardware graph for embedding K n Suppose degree(physical qubit) D (e.g. D = ) Each logical qubit (vertex) of K n will require at least n D physical qubits. That is, hardware graph will need to have Ω(n /D) physical qubits.

28 Triad: Optimal Hardware Graph for Embedding K n US Patent Application US00/00, July, 00. Triad: Satisfy all known physical constraints Admit efficient embedding of any n-vertex graph Optimal for embedding complete graph K n

29 Construction of Triad Each vertex i of K n is mapped to a chain of n virtual vertices Incrementally construct K i+ from K i :

30

31 Decomposition Degree D = Chop the n virtual vertices into n D physical qubits Decomposition: K = K [..] K [..],[..] K [..] K [..],[..] K [..] K [..]

32 Decomposition Degree D = Chop the n virtual vertices into n D physical qubits Decomposition: K = K [..] K [..],[..] K [..] K [..],[..] K [..] K [..]

33 Triad: Optimal Hardware Graph for Embedding K n B B B B B B A A A A B: Complete bipartite graph K D D A: Complete graph K D A is a minor of B (B building block) number of physical qubits N = number of couplers D N n D Question: What are the sparse graphs G m for m > n that can be embeded in U N?

34 Adiabatic Quantum Architecture U sparse Design Problem Let F consist of a set of sparse graphs. Design a F-minor-universal graph U sparse that is as small as possible (# qubits+ #couplers ) subject to Physical Constraints: small constant degree, constant length of a qubit/coupler Embedding Constraint: A minor-embedding can be efficiently computed Remark: tree-width of U sparse needs to necessarily large (ω(log n)) for otherwise the dynamic programming over the tree-decomposition of U sparse would be able to solve the problems in O(c tr(usparse) ) time, once the embedding is given. Geometric Expander? Topological vs geometric graph

35 Adiabatic Quantum Architecture U sparse Design Problem Let F consist of a set of sparse graphs. Design a F-minor-universal graph U sparse that is as small as possible (# qubits+ #couplers ) subject to Physical Constraints: small constant degree, constant length of a qubit/coupler Embedding Constraint: A minor-embedding can be efficiently computed Remark: tree-width of U sparse needs to necessarily large (ω(log n)) for otherwise the dynamic programming over the tree-decomposition of U sparse would be able to solve the problems in O(c tr(usparse) ) time, once the embedding is given. Geometric Expander? Topological vs geometric graph

36 Adiabatic Quantum Architecture U sparse Design Problem Let F consist of a set of sparse graphs. Design a F-minor-universal graph U sparse that is as small as possible (# qubits+ #couplers ) subject to Physical Constraints: small constant degree, constant length of a qubit/coupler Embedding Constraint: A minor-embedding can be efficiently computed Remark: tree-width of U sparse needs to necessarily large (ω(log n)) for otherwise the dynamic programming over the tree-decomposition of U sparse would be able to solve the problems in O(c tr(usparse) ) time, once the embedding is given. Geometric Expander? Topological vs geometric graph

37 Related Work Forbidden minor: (Robertson and Seymour) Given a fixed graph G, (e.g. K ), find a minor-embedding of G in U in O( E(U) ) time (exponential in the size of G). Short paths in expander graphs: (Kleinberg & Rubinfeld, FOCS9) a randomized polynomial algorithm, based on a random walk, to find a minor-embedding in a given degree-bounded expander. Parallel architecture design: (L.F. Thomson. Introduction to parallel algorithms and architectures: arrays, trees, hypercubes.) Different physical constraints: load: # logical qubits maps to a single physical qubit dilation: # of stretched edges (through intermedia qubits) Universal graph design: References in (Alon and Capalbo, SODA 0) Optimal universal graphs with deterministic embedding.

38 Related Work Forbidden minor: (Robertson and Seymour) Given a fixed graph G, (e.g. K ), find a minor-embedding of G in U in O( E(U) ) time (exponential in the size of G). Short paths in expander graphs: (Kleinberg & Rubinfeld, FOCS9) a randomized polynomial algorithm, based on a random walk, to find a minor-embedding in a given degree-bounded expander. Parallel architecture design: (L.F. Thomson. Introduction to parallel algorithms and architectures: arrays, trees, hypercubes.) Different physical constraints: load: # logical qubits maps to a single physical qubit dilation: # of stretched edges (through intermedia qubits) Universal graph design: References in (Alon and Capalbo, SODA 0) Optimal universal graphs with deterministic embedding.

39 Related Work Forbidden minor: (Robertson and Seymour) Given a fixed graph G, (e.g. K ), find a minor-embedding of G in U in O( E(U) ) time (exponential in the size of G). Short paths in expander graphs: (Kleinberg & Rubinfeld, FOCS9) a randomized polynomial algorithm, based on a random walk, to find a minor-embedding in a given degree-bounded expander. Parallel architecture design: (L.F. Thomson. Introduction to parallel algorithms and architectures: arrays, trees, hypercubes.) Different physical constraints: load: # logical qubits maps to a single physical qubit dilation: # of stretched edges (through intermedia qubits) Universal graph design: References in (Alon and Capalbo, SODA 0) Optimal universal graphs with deterministic embedding.

40 Related Work Forbidden minor: (Robertson and Seymour) Given a fixed graph G, (e.g. K ), find a minor-embedding of G in U in O( E(U) ) time (exponential in the size of G). Short paths in expander graphs: (Kleinberg & Rubinfeld, FOCS9) a randomized polynomial algorithm, based on a random walk, to find a minor-embedding in a given degree-bounded expander. Parallel architecture design: (L.F. Thomson. Introduction to parallel algorithms and architectures: arrays, trees, hypercubes.) Different physical constraints: load: # logical qubits maps to a single physical qubit dilation: # of stretched edges (through intermedia qubits) Universal graph design: References in (Alon and Capalbo, SODA 0) Optimal universal graphs with deterministic embedding.

41 Theorem (A Tighter Bound) (For chains topological-minor-embedding) ( def For i V(G), C i = j nbr(i) J ij h i ). Set { h i j k = sign(h i ) τ(j,i) Onbr(i k ) J ij C i / j τ(j,i) Onbr(i k ) J ij i k is a leaf of T i otherwise and F e i < C i /, for each e E(T i ). Then there is one-one correspondence between arg min E and arg min E emb. V. Choi. Minor-embedding in adiabatic quantum computation: I. The parameter setting problem. Quantum Inf. Processing.,, 9 09, 00.

42 QUBO Ising Model J i J i c i J i h i = ( J ij c i ) J i J id J id embed (J i + J i ) c i Ji 0 F F J i J i J i (J i(d ) + J id ) ci (J>0) F < c i

43 Sketch of the Proof. Case: For h > 0, J > 0 h + F J + J + h J + J + F E (h J + J + F ) ( )(h J + J + F ) = (h J + J + F ) Therefore if F < h (J + + J + ) = ( J C/) J = C/, E > 0.

44 Case: h > 0, Sketch of the Proof (Cont.) h + F J + J h F J + J E (h J + +J F ) ( )(h J + +J F ) = (h J + +J F ) Therefore if F < h (J + + J ) = ( J C/) J = C/, E > 0.

45 Ising Hamiltonian for Solving MIS Recall Min E(s,..., s n ) = h i s i + J ij s i s j i V(G) ij E(G) where h i = j nbr(i) J ij, for i V(G), J ij >. Set J ij = J = + ɛ, for some ɛ > 0. Embedded Ising Hamiltonian for Solving MIS C i = j nbr(i) J ij h i = F i < C i / =, set F i = ( + ɛ) h i {dj, (d )(J ), (d )J : d < deg i} Example deg i =, {h, J, F } = { ( + ɛ), 0, ɛ, + ɛ, + ɛ}

46 Conclusion Minor-embedding in adiabatic quantum optimization Adiabatic quantum architecture design Some Research Problems: Measure of goodness of a minor-embedding Efficience of minor-embedding reduction: how does minor-embedding effect the min-spectral gap? What are the classically hard problems on sparse graphs? Sparse-graph-minor universal architecture design problem V. Choi. Minor-embedding in adiabatic quantum computation: I. The parameter setting problem. Quantum Inf. Processing.,, 9 09, 00. V. Choi. Minor-embedding in adiabatic quantum computation: II. Minor-universal graph design. To appear in arxiv.

47 Acknowledgement D-Wave Systems Inc. Mohammad Amin Andrew Berkley Mark Johnson Jan Johannson Richard Harris Trevor Lanting Paul Bunyk Felix Maibaum Fred Brito Colin Truncik Andy Wan Fabian Chudak Bill Macready Geordie Rose David Kirkpatrick (UBC) Robert Rossendorf (UBC) Bill Kaminsky (MIT) Peter Young (UC Santa Cruz) Siyuan Han (U. of Kansas)