Introduction to Adiabatic Quantum Computation

Introduction to Adiabatic Quantum Computation Vicky Choi Department of Computer Science Virginia Tech April 6, 2

Outline Motivation: Maximum Independent Set(MIS) Problem vs Ising Problem 2 Basics: Quantum Computation 3 Adiabatic Quantum Computation Problem Hamiltonian Initial Hamiltonian Adiabatic Running Time 4 Minor-embedding Adiabatic Quantum Architecture Design

Maximum Independent Set Problem Input: a graph G, V(G) = {, 2,..., n}, Output: mis(g) V(G), independent, mis(g) is maximized independent: for i, j mis(g), ij E(G) Example 2 3 5 4 6

Formulated as a Quadratic Binary Optimization Problem For i V(G), associate it with a binary variable x i {, }. Define Y(x,..., x n ) = x i J ij x i x j. quadratic pseudo-boolean function i V(G) (i,j) E(G) Example: for J ij = 2 Y(,,,,, ) = 2 Y(,,,,, ) = + 2 + = 2 3 5 4 6

Formulated as a Quadratic Binary Optimization Problem For i V(G), associate it with a binary variable x i {, }. Define Y(x,..., x n ) = i V(G) quadratic pseudo-boolean function Theorem If J ij > for all ij E(G), then x i (i,j) E(G) mis(g) = max Y(x,..., x n ). and mis(g) = {i V(G) : xi = }, where (x,..., x n ) = argmax (x,...,x n) {,} ny(x,..., x n ). J ij x i x j.

Energy Function of the Ising Model E(s,..., s n ) = i V(G) h i s i + where spin s i {, +}, h i, J ij R. Ising Problem ij E(G) J ij s i s j Find the configuration of spins such that E(s,..., s n ) is minimized.

Correspondence x i = + s i 2 x i = s i =, x i = s i = + is equivalent to Max Y(x,..., x n ) = x i J ij x i x j i V(G) ij E(G) 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 2, for i V(G).

Ising Problem: Naturally solved by an adiabatic quantum processor

What is a Qubit? classical bit: or quantum bit (qubit): a unit vector in C 2 Dirac notation: (ket) p = a + b def = ( ) def, = ( ) a 2 + b 2 =, a, b C (complex number) SUPERPOSITION of and

Measurement observe with probability a 2 observe with probability b 2 After measurement, the state is or Measurement changes the state! Example p = 2 ( + ) Prob[] = /2 Prob[] = /2

Two Qubits 2 classical bits: {,,, } 2 qubits: a unit vector in C 22 = = ( ) ( ) = ( ) ( ) = ( ) = = ( ) ( ) = ( ) = = ( ) ( ) = ( ) = = ( ) ( ) = ( ) ψ = a + a + a 2 + a 3, x {,} 2 a x 2 = Prob[x {, } 2 ] = a x 2

n Qubits n qubits : a unit vector in C 2n, superposition of 2 n possible states {...,...,...,... } ψ = x {,} n α x x, x {,} n α x 2 = x = x x 2... x n = x... x n Example (Uniform superposition) Prob[x {, } n ] = 2 n ψ = 2 n x i {, }, i =,..., n x {,} n x

Quantum Evolution (Axiom) All quantum systems evolve in time according to the Schrödinger equation: i d ψ(t) = H(t) ψ(t) dt The dynamic of a quantum system is completely determined by its time-dependent Hamiltonian H(t). ψ(t) : the state of the n-qubit quantum system at time t Hamiltonian: 2 n 2 n (complex) matrix Hermitian: H = H (all eigenvalues are real) Eigenvalues of the Hamiltonian = energy levels of the system (eigenvectors/eigenstates) Ground state of H: eigenstate of H with lowest eigenvalue/energy

Adiabatic Quantum Computation (AQC) Introduced by Farhi et al in 2 [Science 292(2):472 476, 2] To solve NP-hard optimization problem Example: 3-SAT AQC vs Conventional Quantum Computation AQC is equivalent to the quantum circuit model (Aharonov et al FOCS24) Solve the factoring problem in polynomial time (Shor s algorithm) (will break the RSA public key cryptosystem, including all current internet transactions)

System Hamiltonian: Adiabatic Quantum Algorithm H(t) = ( s(t))h init + s(t)h problem for t [, T ], s() =, s(t ) =. Initial Hamiltonian: H() = H init ground state known (easy to construct) 2 Problem Hamiltonian: H(T ) = H problem ground state encodes the answer to the desired optimization problem 3 Evolution path: s : [, T ] [, ], 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 problem!

Problem Hamiltonian H problem for Ising Problem Recall Ising Problem: Min E(s,..., s n ) = h i s i + J ij s i s j i V(G) ij E(G) Claim: E(s,..., s n ) is the energy function of Ising Hamiltonian: H Ising = i V(G) h i σ z i + ij E(G) J ij σ z i σ z j. Set H problem = H Ising, ground state of H problem minimizes E(s,..., s n )

Pauli Matrices [ ] σ z = σ x = [ ] Eigenvalues and Eigenvectors of σ z I = [ ] σ y [ ] [ ] [ ] [ ] [ ] = [ ] = ( ) Dirac notation: σ z =, σ z = In short: σ z x = ( ) x x, x {, }

Tensor Products [ ] [ ] a a A = 2 b b B = 2 a 2 a 22 b 2 b 22 [ ] A B def a B a = 2 B a 2 B a 22 B dim(a B) = dim(a)dim(b) Tensor product property: (A B)( u v ) = A u B v

Define σ z i, σ z i σ z j σ z i def = I I... σ z... I (σ z is in the ith term) Similarly, σ z i σ z j def = I... σ z... σ z... I (σ z s are in the ith, jth terms) For x = x... x n, x i {, }, i =,..., n. σi z x = (I... σ z... I)( x... x i... x n ) = x... ( ) x i x i... x n = ( ) x i x

H Ising x = h i σi z + J ij σi x σj z x i V(G) ij E(G) = h i ( ) x i + J ij ( ) x i ( ) x j x i V(G) ij E(G) Therefore, the energy function of H Ising is E(x,..., x n ) = h i ( ) x i + J ij ( ) x i ( ) x j i V(G) ij E(G) where x i {, }, for i =,..., n. Replace ( ) x i by s i {+, } E(s,..., s n ) = h i s i + J ij s i s j i V(G) ij E(G)

Adiabatic Quantum Algorithm H(t) = ( s(t))h init + s(t)h problem for t [, T ], s() =, s(t ) =. Initial Hamiltonian: H() = H init ground state known (easy to construct) 2 Problem Hamiltonian: H(T ) = H problem ground state encodes the answer to the desired optimization problem 3 Evolution path: s : [, T ] [, ], e.g., s(t) = t T T : running time of the algorithm

Recall: Initial Hamiltonian H init H init = σ x = eigenvalue i V(G) [ ] σ x i eigenvector + 2 2 ground state of H init is the uniform superposition : = + +... 2 2 2 n x {,} n x

Adiabatic Theorem: Running time vs Spectral gap For s(t) = t/t. If T is large enough: T = O where minimum spectral gap gap min = ( ) poly(n) gap 2 min min (E (t) E (t)), t T E (t) < E (t) <... are the energy levels of H(t). Then the system remains close to the ground state of H(t).

Example: Engergy Spectrum

Adiabatic Theorem: Running time vs Spectral gap For s(t) = t/t. If T is large enough: T = O ( ) poly(n) gap 2 min the system remains close to the ground state of H(t). Note: /gap min is polynomial (exponential resp.), T is polynomial (exponential resp.). What s the accurate (tight) bound for T? Controvercy about adiabatic theorem (quantitative)...

What s the adiabatic running time for MIS/Ising Problem? 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) Decomposed State Evolution Visualization (DeSEV): unveil the quantum evolution blackbox

DeSEV: An Example

Problem Hamiltonian H problem for Ising Problem Recall 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 problem = h i σi z + J ij σi z σj z. G : hardware graph i V(G) ij E(G)

Input Graph (G) vs Hardware Graph (U) Input Graph G Hardware Graph (U) 4 8 3 7 2 6 5 Physical Qubit & Coupler What if G is NOT a subgraph of the hardware graph U?

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). 4 8 3 3 4 8 3 7 2 7 2 6 6 5 5 G φ(g) = G emb

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 )

Why does the minor-embedding work? 4 8 3 3 4 8 3 7 2 7 2 6 6 5 5 Ferromagnetic(FM)-coupling: F < 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? V. Choi. Minor-embedding in adiabatic quantum computation: I. The parameter setting problem.

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.

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. 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 V. Choi. V. Choi. Minor-Embedding in Adiabatic Quantum Computation: II. Minor-Universal Design Problem. arxiv:quant-ph/.36, 2.

Quantum Processor that Implements Ising Hamiltonian H(t) = h i (t)σi z + J ij (t)σi z σj z + i (t)σi x i V(G) ij E(G) i V(G) Initial Hamiltonian: H init = i V(G) iσ x i Problem Hamiltonian: H problem = i V(G) h i σ z i + ij E(G) J ij σ z i σ z j. Energy function of H problem E(s,..., s n ) = h i s i + J ij s i s j i V(G) ij E(G)

Summary AQC is equivalent to conventional quantum computation (Aharonov et al FOCS24) a new paradigm for directly attacking NP-hard optimization problem New challenges to computer scientists: Continuous algorithms Fundamental algorithmic problems in AQC Spectral gap analysis How to design a good initial Hamiltonian How to best encode the computational problem into a problem Hamiltonian How to design an efficient evolution path V. Choi. Adiabatic Quantum Algorithms for the NP-Complete MIS, Exact Cover and 3SAT Problems. arxiv:quant-ph:4.2226, 2.

Acknowledgement D-Wave Systems Inc. David Kirkpatrick (UBC) Robert Rossendorf (UBC) Bill Kaminsky (MIT) Peter Young (UC Santa Cruz) Siyuan Han (U. of Kansas)