Numerical Studies of Adiabatic Quantum Computation applied to Optimization and Graph Isomorphism

Numerical Studies of Adiabatic Quantum Computation applied to Optimization and Graph Isomorphism A.P. Young http://physics.ucsc.edu/~peter Work supported by Talk at AQC 2013, March 8, 2013 Collaborators: I. Hen, E. Farhi, P. Shor, D. Gosset, A. Sandvik, V. Smelyanskiy, S. Knysh, M. Guidetti.

Plan Quantum Monte Carlo (the method used to study large sizes) Results for SAT-type problems Results for a spin-glass problem A first look at graph isomorphism

Quantum Adiabatic Algorithm Proposed by Farhi et. al (2001) to solve hard optimization problems on a quantum computer. Want to find the ground state of a problem Hamiltonian H P involving Ising spins, z, or equivalently, bits i = ±1 b i =0or 1 Make quantum by adding a non-commuting driver Hamiltonian. Simplest is a transverse field: H D = Total Hamiltonian: H(t) = [1 with s(0) = 0, s(τ) = 1. h NX i=1 x i 1 s(t)]h D + s(t)h P H D (g.s.) adiabatic? H P (g.s.?) s 0 1 System starts in ground state of driver Hamiltonian. If process is adiabatic (and T 0), it ends in g.s. of problem Hamiltonian, T

Quantum Monte Carlo Early numerics, Farhi et al. for very small sizes N 20, on a particular problem found the time varied only as N 2, i.e. polynomial! But possible crossover to exponential at larger sizes? Want to estimate how the running time varies with size for large sizes. We therefore have to do a Quantum Monte Carlo Sampling of the 2 N states. QMC can efficiently study only equilibrium properties of a quantum system by simulating a classical model with an extra dimension, imaginary time, τ, where 0 <1/T. Used a version called the stochastic series expansion (SSE), pioneered by Sandvik. Not perfect, (statistical errors, need to ensure equilibration) but the only numerical method available for large N.

Quantum Phase Transition H D H P 0 QPT 1 s Bottleneck is likely to be a quantum phase transition (QPT) where the gap to the first excited state is very small E E 1 E 0 E min Landau Zener Theory: To stay in the ground state the time needed is proportional to E 2 min s

Examples of results with the SSE code Time dependent correlation functions decay with τ as a sum of exponentials <H p ()H p (0)> - <H p > 2 A( )A(0) A 2 = n=0 0 A n 2 exp[ (E n E 0 ) ] For large τ only first excited state contributes, pure exponential decay 10 1 0.1 0.01 0.001 Text 3-reg MAX-2-XORSAT, N=24 diag. QMC = 64, symmetric levels 0 5 10 15 20 25 30 Small size, N= 24, excellent agreement with diagonalization. C() 1 0.1 0.01 0.001 Text 3-reg MAX-2-XORSAT, N = 128 QMC fit, E = 0.090 0 10 20 30 40 50 Large size, N = 128, good quality data, slope of straight line gap.

Dependence of gap on s Text Text Results for the dependence of the gap to the first excited state, ΔE, with s, for one instance Text of 1-in-3 SAT with N = 64. The gap has a minimum for s about 0.66 which is the bottleneck for the QAA. s

Dependence of gap on s Text s Mind THIS gap Text Results for the dependence of the gap to the first excited state, ΔE, with s, for one instance Text of 1-in-3 SAT with N = 64. The gap has a minimum for s about 0.66 which is the bottleneck for the QAA. We compute the minimum gap for many (50) instances for each size N and look how the median minimum gap varies with size.

K-SAT (like) with K > 2 Results for several K-SAT(like) problems for K > 2: Locked 1-in-3 SAT: The clause is formed from 3 bits picked at random. The clause is satisfied (has energy 0) if one is 1 and the other two are 0. Locked 2-in-4 SAT 3-regular 3-XORSAT (3-spin model) 3-regular means that each bit is in exactly three clauses. 3-XORSAT means that the clause is satisfied if the sum of the bits (mod 2) is a value specified (0 or 1) for each clause. In terms of spins σ z (= ±1) we require that the product of the three σ z s in a clause is specified (+1 or -1). Found minimum gap varies exponentially with N. The bottleneck appears to be a very small gap at the quantum phase transition. Show this in the next two slides.

median E min 0.1 0.01 Locked 1-in-3 Plots of the median minimum gap (average over 50 instances) 2 / ndf = 1.35 Q = 0.26 median E min 0.1 0.01 2 / ndf = 18.73 Q = 3.82e-12 0.16 exp(-0.042 N) 0.001 10 20 30 40 50 60 70 80 90 100 N Exponential fit 5.6 N -1.51 10 100 N Power law fit Clearly the behavior of the minimum gap is exponential

Curious that the hardest problem for these heuristic algorithms is the one with a polynomial time algorithm (complexity class P). Comparison with a classical algorithm, WalkSAT: QAA WalkSAT 10 9 0.1 10 8 10 7 median E min 0.01 median flips 10 6 10 5 10 4 0.001 1-in-3, 0.16 exp(-0.042 N) 2-in-4, 0.32 exp(-0.063 N) 3-XORSAT, 0.22 exp(-0.080 N) 10 20 30 40 50 60 70 80 90 100 N 10 3 10 2 10 1 3-XORSAT, 158 exp(0.120 N) Exponential behavior for both QAA and WalkSAT 2-in-4, 96 exp(0.086 N) 1-in-3, 496 exp(0.050 N) 0 50 100 150 200 250 300 The trend is the same in both QAA and WalkSAT. 3-XORSAT (a 3-spin Hamiltonian) is the hardest, and locked 1-in-3 SAT the easiest. N

Spin glass problem (2-spin clauses ) Found minimum gap at the quantum phase transition, s = s*, varies polynomially with N. However, the gap in the spin glass phase, s > s*, appears to vary exponentially with N. The bottleneck is therefore in the spin glass phase, not at the quantum critical point. Will show this in the next few slides.

Spin glass on a random graph: 3-Reg Max-Cut: Max means we are in the UNSAT phase, and want to find the bit string with the least number of unsatisfied clauses. In Max-Cut, each clause has two bits and we want to maximize the number of clauses in which the bits are different. In physics jargon this is an anti-ferromagnet. Replica theory indicates that these 2-SAT-like problems are different from K-SAT problems for K > 2. (Hence we study it here.) 3-Regular means that it bit is in three clauses, i.e. has 3 neighbors connected to it. The connected pairs are chosen at random. 1 3 i Note: there are large loops 2

3-Reg MAX-Cut: II The problem Hamiltonian is H P = 1 2 i,j 1+ z i z j Note the symmetry under z i Cannot form an up-down antiferromagnet because of loops of odd length. In fact, it is a spin glass. Adding the driver Hamiltonian there is a quantum phase transition at s = s * above which the symmetry is spontaneously broken. Cavity calculations (Gosset, Zamponi) find s * 0.36 Investigated the problem near s * (s 0.5). Also just considered instances with a unique satisfying assignment (apart from the degenerate state related by flipping all the spins). (These are exponentially rare.) z i, i

3-Reg MAX-Cut: IV (Gap) E 1 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0.35 0.4 0.45 0.5 s For larger sizes, a fraction of instances have two minima, one fairly close to s * ( 0.36) and other at larger s in the spin glass phase. Figure shows an example for N = 128. Hence did 2 analyses (i) Global minimum in range (up to s=0.5) (ii) If two minima, just take the local minimum near s *. E min 0.3 0.2 0.1 N 10 20 50 100 5.31 N -0.96 0.29 exp(-0.014 N) 20 40 60 80 100 120 140 160 180 N 0.3 0.2 0.1 0.05 E min Global [exponential (main figure) preferred over E min 0.3 0.2 0.1 power-law (inset)] Local [power-law best (inset)] N 10 20 50 100 2.79 N -0.78 minimum gap near s=0.36 0.26 exp(-0.011 N) 20 40 60 80 100 120 140 160 180 N 0.3 0.2 0.1 0.05 E min

Graph Isomorphism I Are two graphs the same upon permuting the indices? Hypothesis: All non-isomorphic graphs can be distinguished by putting a suitable Hamiltonian on the edges of a graph, running the QAA, and computing appropriate average quantities, not necessarily at the end of the algorithm, by repeated measurements. So far: have tested for some small graphs N 29 (works). Could be tested experimentally on D-Wave One (128 nodes). We tried an antiferromagnet H P (G) = X hi,ji2g Main results for Strongly Regular Graphs. Families of similar but non-isomorphic graphs are known. We considered sizes from N = 15 to 29 vertices. Computed, e.g. M x 1 N X h i x i, i Q 2 z i 1 z j N(N 1) X i6=j h z i z j i 2

0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.05 0.10 0.15 0.20 0.25 adiabatic parameter s spinglass order parameter Q2 Instance 2 Instance 1 4 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.05 0.10 0.15 0.20 0.25 adiabatic parameter s spinglass order parameter Q2 Instance 2 Instance 1 FIG. 1: Spin-glass order parameter Q 2 in the ground state for the two non-isomorphic strongly-regular graphs (SRGs) on N =16vertices. Asthefigureindicates,thetwographs show different Q 2 values starting from s 0.4. matrix has only three distinct values [35]: λ 0 = k, which is non-degenerate, and λ 1,2 = 1 2 ( λ µ ± N ), which are both highly degenerate. Both the value and degeneracy of these eigenvalues depend only on the family parameters, so within a particular SRG family, all graphs are cospectral [35]. These highly degenerate spectra are one reason why distinguishing non-isomorphic SRGs is difficult. 1.88 1.90 1.92 1.94 1.96 1.98 2.00 0.3 0.4 0.5 0.6 0.7 0.8 0.9 M x Q2 6.2 6.3 6.4 6.5 6.6 6.7 6.8 0.16 0.18 0.20 0.22 0.24 0.26 M x Q2 FIG. 2: Scatterplot in the M x Q 2 plane of the 15 strongly regular graphs with N =25verticesinthelimits 1. The horizontal axis is the magnetization along the x-direction and the vertical axis is the spin-glass order parameter Q 2. The circled data point corresponds to the only two graphs which are not distinguished by values of Q 2 and M x in the s 1limit. Thesetwoinstancesare,nonetheless,distinguished by measurements at s<1 (see text). The inset shows the M x Q 2 values of two of the graphs that lie outside of the region shown in the main panel. perturbation theory suffices to distinguish between all but two of the graphs (the latter two share the same Q 2 Graph Isomorphism II The spin glass order parameter Q2 against s for two Strongly Regular Graphs with N = 16 vertices. They are clearly distinguished. Scatter plot of Q2 against Mx in the s 1 limit for the 15 SRG s with N=25. All but two are distinguished, and these two turn out to be distinguished at intermediate values of s. On D-Wave One would need to consider different, but related quantities. Could also look at energy distribution as a function of runtime (M. Amin). Also look at distributions (Farhi), e.g. rather than Hen and Young, Phys. Rev. A 86, 042310 (2012) P ( c M x ) Diagonalization & degenerate pert. theory M x h c M x i

Graph Isomorphism III (other graphs) Q 2 10 4 0.04 0.03 0.02 0.01 0.00 0.01 0.1 0.05 0 0 0.5 1 s E 10 5 0.0 0.2 0.4 0.6 0.8 1.0 adiabatic parameter s Other graphs, FIG. 5: The e.g. difference two N in = Q14 2 values, graphs Qof 2 (main Emms panel) et al, andchosen to be particularly energies had (inset), to distinguish. for the two graphs QAA Gworks, 1 and G 2 at with least N =14 in principle, differences that certain non-zero, quantum but walkers small, cannot in the distinguish vicinity [27, of 28]. minimum gap. Longer term, need to investigate size-dependence of minimum gap. B. Other pairs of graphs

Conclusions

Conclusions Simple application of QAA seems to fail:

Conclusions Simple application of QAA seems to fail: At the (first order transition) for SAT problems

Conclusions Simple application of QAA seems to fail: At the (first order transition) for SAT problems In the spin glass phase, beyond the transition, in the spin glass model

Conclusions Simple application of QAA seems to fail: At the (first order transition) for SAT problems In the spin glass phase, beyond the transition, in the spin glass model What about other models?

Conclusions Simple application of QAA seems to fail: At the (first order transition) for SAT problems In the spin glass phase, beyond the transition, in the spin glass model What about other models? Perhaps the QAA could be useful to get an approximate ground state. (For some problems, even this is difficult). I have just described the simplest implementation of the QAA. Perhaps if one could find clever paths in Hamiltonian space one could do better. A subject for future research.