Application of graph theory in quantum computer science

Adam Glos

Institute of Theoretical and Applied Informatics,
Polish Academy of Sciences

Gliwice, 15.09.2021

Random and quantum walks

• A model for physical processes (like Brownian motion)
• A model for heuristic optimization algorithms (simulated annealing, tabu search, etc.)
• Can be used for the analysis of social structures
• Toy model for quantum computing, recently used for quantum optimization

Thesis

[1] A. M. Childs and J. Goldstone, “Spatial search by quantum walk,” Physical Review A, vol. 70, no. 2, p. 022314, 2004.

The continuous-time quantum walk models remain powerful for nontrivial graph structure

Fast propagation of time-independent CTQW

• \(p_k(t)\) - probability of being in \(k\) after time \(t\)
• \(\mu_2(t) = \sum_{k\in Z} k^2 p_k(t)\) - mean square distance from the initial point
• \(\mu_2(t) = \Theta(t^{\alpha \pm \epsilon})\), \(\alpha=2\) means the walk is fast (like CTQW), \(\alpha=1\) means slow walk (like random walk)

0

1

3

-1

-2

-3

2

• No \(L\):  CTQW, undirected graphs, fast propagation
• \(L = |v\rangle\langle u|\) for \((u,v)\in \vec E\), local environment interaction Quantum Stochastic Walk (LQSW), slow propagation, any directed graph

GKSL master equation:

directed

undirected

Fast propagation of time-independent CTQW

Theorem (dissertation)

• For \(\omega =1\):  slow propagation (like in random walk)
• For \(\omega < 1\):  fast propagation (like in CTQW)

Fast propagation of time-independent CTQW

\(L=\sum_{(u,v)\in \vec E} |v\rangle \langle u|\), global environment interaction QSW)

Conclusion: We designed fast quantum walk, even for weak Hamiltonian impact (large for \(\omega\)).

Fast propagation of time-independent CTQW

Fit function: \(p_1 - p_2/(p_3-t)^{p_4}\)

\(p_s\): the probability of being in the sink

\(\mu_s\): second-order moment of distance from the sink

Conclusion: NMQSW converges better for stronger Hamiltonian interactions than LQSW

Comment: For NMQSW, \(p_s\) is sometimes below 0.99 (6% cases)

Fast propagation of time-independent CTQW

Fast propagation of time-independent CTQW

Simulated annealing

• Random walk over graph of solutions
• The longer the computation, the more favor smaller energy
• Mixture of (time-dependent) undirected and directed graph evolution!

directed

undirected

CTQW spatial search for heterogeneous graphs

• \(w\) is the marked vertex
• \(M_G\) is the graph matrix
• We start in a special, easy-to-prepare state
• \(\gamma\) has to be adjusted for \((w, M_G)\)

My results (adjacency matrix):

• We corrected the proof of the above
• We relaxed the condition on \(p\) into \(p \gg \log n/n\)
• For \(p  \gg \log^3(n)/(n\log^2\log(n))\) all vertices can be found almost surely in \(O(\sqrt{n})\)

Theorem [1]. Let \(G(n,p)\) be random Erdős-Rényi model and \(M_G\) be an adjacency matrix. If \(p\geq \log^{3/2}(n)/n\), then we almost surely can find almost all vertices in \(O(\sqrt{n})\) time

[1] Chakraborty, S., Novo, L., Ambainis, A., & Omar, Y. (2016). Spatial search by quantum walk is optimal for almost all graphs. Physical review letters, 116(10), 100501.

CTQW spatial search for heterogeneous graphs

My results (Laplacian):

• For \(p \gg\log n/n\) all vertices can be found in \(O(\sqrt{n})\)
• For \(p=p_0 \log n/n\) for \(p_0>1\) all vertices can be found in \(O(\sqrt{n})\) with \(\Omega(1)\) probability

CTQW spatial search for heterogeneous graphs

• Requires good knowledge about the eigendecomposition of \(M_G\)
• For adjacency matrix \(A\), it is very robust
• For Laplacian \(L\coloneqq D-A\), if the deviation between the smallest and the largest degrees is large, there is no spectral gap

• There are known theorems with weaker conditions, but difficult to use

CTQW spatial search for heterogeneous graphs

Normalized Laplacian \(M_G= I - D^{-1/2} AD^{-1/2}\)

• Classical search: \(\tilde\Theta(nn^{ b(1-\frac{i}{n})})\)
• Quantum search with adjacency matrix: \(\tilde \Theta(\sqrt{n}n^{ b(1-\frac{i}{n})}\))

• Quantum search with normalized Laplacian: \(\tilde\Theta(\sqrt{nn^{ b(1-\frac{i}{n})}}\))

   

Chung-Lu with expected degree \(\omega_i = n^{a+\frac{i}{n} b}\) for \(i\)-th vertex

CTQW spatial search for heterogeneous graphs

Theorem (dissertation) if there is spectral gap for normalized Laplacian, then we observe the quadratic speed-up over classical search

Barabási-Albert (designed as in [1])

highest degree

smallest degree

[1] Chakraborty, S., Novo, L., & Roland, J. (2020). Optimality of spatial search via continuous-time quantum walks. Physical Review A, 102(3), 032214.

CTQW spatial search for heterogeneous graphs

Normalized Laplacian \(M_G= I - D^{-1/2} AD^{-1/2}\)

Theorem (dissertation) if there is spectral gap for normalized Laplacian, then we observe the quadratic speed-up over classical search

Q: What if we don't know optimal measurement time?

The procedure:

1. let \(\mathcal A(t)\) be the quantum search with measurement time \(t\)
2. choose \(C,K'>0\) arbitrarily
3. Let \(K = K'\log n\)
4. run \(\mathcal A\)(t) with times \(t = C, Cn^{\frac{1}{K}},Cn^{\frac{2}{K}},\ldots,Cn^{2}\)
5. Stop when the marked vertex is found

Hypothesis: Under some assumptions on \(\mathcal A\), the procedure above has the same time complexity as the optimal one

CTQW spatial search for heterogeneous graphs

I would like to thank Prof. Łuczak for his positive review.

Q #1: In the first part of the thesis a very elegant generalization of the definition of a CTQW is presented for directed graphs. It was not clear to me, whether there are some interesting applications where one could use these walks?

Answer:

1. NGQSW could be used for an alternative quantum version of simulated annealing
2. GQSW could be used for multiplex graphs quantum walks

Q #2: The applicant considers various graphs, I found especially interesting the case of complex, Barabási-Albert type graphs. Would it be possible to apply or generalize some of the presented ideas to graphs representing fractal structures, like the Sierpiński triangle?

Answer:

1. It likely depends on the structure of the graph
2. For the Sierpiński graph, the conditions are not satisfied for any graph matrix chosen
3. The measurement time still does not have to be known (under assumptions from dissertation)

Q #3: Is there some fundamental reason behind the difference between the Laplacian vs adjacency matrix behavior? Are these the only interesting possibilities to construct the dynamics or one could search for other matrices as well?