Quantum Complexity Theory

1. Quantum Complexity Theory • Heuristic methods have been developed to solve classical problems (simplex method for linear programming, simulated annealing, Monte Carlo methods). • Classical complexity theory aims at making rigorous what problems can be solved efficiently and which ones are hard (and how hard they are). • Physicists have developed (heuristic) methods to solve quantum problems (perturbation theory, WKB approximation, mean-field theory etc. etc.) • Quantum complexity theory aims at showing which problems a classical machine can solve efficiently, which ones a quantum computer can do better, and which ones are hard and how hard. • The how hard-aspect is sometimes characterized in terms of interaction of an ordinary human being (Arthur) with an all-powerful prover (Merlin).

2. NP Success story of complexity theory in physics: the notion of NP and NP-completeness. Is the question PNP a mathematics or physics question: A mathematical question with physical consequences, a universe where P=NP would probably be much different. Useful notion of NP-completeness.

3. Satisfiability The k-SAT problem: Let x be an n-bit string. Let Ci be a constraint that depends only on the value of some k bits of x. A bitstring x can either satisfy or not satisfy the constraint Ci depending on the value of these k bits. Take a set of constraints Ci for i=1 to m with m=poly(n). Q: Is there a bit string that satisfies all m constraints? If yes, a prover can give you the bitstring and you can check the m constraints. If no, no prover can convince you that there is such bitstring. k-SAT is NP-complete

4. Beyond NP What is the physical relevance of these classes? What are the ‘complete’-problems? QMA (or Quantum NP) NP MA Stoquastic MA MA ~probabilistic version of NP QMA ~quantum version of NP

5. Definitions

6. The Story • The Quantum Ising Spin Glass problem is complete for QMA. • Note that QMA allows for an error if the answer is yes (if yes, Arthur decides yes with probability at least (say) 2/3). Note that NP and MA • do not have this error (MA allows for error in no-instances) • For the class QMA1 in which Arthur decides Yes with probability 1, • the complete problem is the quantum satisfiability problem. • For MA the complete problem (first known MA-complete problem) is the stoquastic satisfiability problem. Relation to stoquastic Hamiltonians and classical satisfiability problem. • For stoquastic MA (weird class) the complete problem is the • Stoquastic Ising Spin Glass problem. • Beliefs in computer science: MA=NP=Stoquastic MA=AM

7. Quantum Ising Spin Glass Quantum Ising Spin Glass (QISP): Given is a k-local Hamiltonian H=iHi on n qubits and given  <  (with |-|  1/poly(n)). ||Hi||  poly(n). We are promised that the ground-state energy (H)) is either   or  . Q: Is there a state with energy less than ? k-local means that the Hamiltonian is a sum of terms each of which acts non-trivially on at most k qubits. Kitaev: 5-local QISP is QMA-complete. Strongest results to date: 2-local QISP on qudits of a 1-D chain is QMA-complete (Aharonov,Gottesman, Kempe, Irani). Quantum Ising Spin glass is also called the local Hamiltonian problem.

8. Quantum k-SAT Ground-state of Hamiltonian is not always the one which is the ground-state of each individual term in the Hamiltonian. `frustration free property’ So a different question to askis: is there a state which is the ground-state of each local term in a quantum Ising spin glass Hamiltonian? Such state satisfies all local constraints. We can always shift the Hamiltonian such that the energy of this satisfying state is 0. Then the satisfying state has the property of being in the null-space of a bunch of k-local projectors…. Quantum k-SAT on n qubits: Given a set of k-local projectors Pi on n qubits and   1/poly(n). Either - there is a state that is in the space spanned by all projectors Pi. (yes-instance), - or any state  there exists an i such that <| Pi |>  1- (no-instance). (Note we are switching null-space and +1 space here…)

9. Connection with classical k-SAT Classical k-SAT problem can be viewed as quantum k-SAT where the projectors are diagonal in computation basis. Some bit-strings in +1 space, some in null-space. Promise on no-instances is not needed. There is something in between diagonal k-local projectors and general k-local projectors….the stoquastic k-SAT problem

10. Stoquastic Hamiltonians Physicists have known for a long time that not all Hamiltonians are created equal. Some are more quantum than other’s. Physics lingo: “Avoid the sign-problem”, “Quantum ground-state problem mapped onto classical partition function problem”, Green’s function Monte Carlo techniques” What is exactly the property of these Hamiltonians and what does this feature do to the complexity of the lowest-eigenvalue problem, the power of these Hamiltonians for implementing an adiabatic quantum computer, etc?

11. Stoquastic Hamiltonians General definition: stoquastic Hamiltonians are real and have non-positive off-diagonal elements in some standard (product) basis |i>. Then G=I-t H for some (real) t is a non-negative matrix (in this basis) or consider the nonnegative Gibbs matrix G=e-bH . Perron-Frobenius: Eigenvector with largest eigenvalue of G (ground-state of H) has the property that |>=i i |i> where i  0. P(i)=i/i i is a probability distribution. Our excuses for the word stoquastic….introduced not without reason.

12. Examples of Stoquastic Hamiltonians Particles in a potential; Hamiltonian is a sum of a diagonal potential term in position |x> and off-diagonal negative kinetic terms (-d2/dx2). All of classical and quantum mechanics. Quantum transverse Ising model Ferromagnetic Heisenberg models (modeling interacting spins on lattices) Jaynes-Cummings Hamiltonian (describing atom-laser interaction), spin-boson model, bosonic Hubbard models, Bose-Einstein condensates etc. D-Wave’s Orion quantum computer… Non-stoquastic are typically fermionic systems, charged particles in a magnetic field. Stoquastic Hamiltonians are ubiquitous in nature. Note that we only consider ground-state properties of these Hamiltonians.

13. Local Stoquastic Hamiltonians • Remember k-local Hamiltonians are those that can be written as a • sum ofterms acting non-trivially on k qubits (qudits). • Our results hold for local term-wise stoquastic Hamiltonians, each • local term is stoquastic. • Side Remark: • Local termwise-stoquastic is not necessarily the same as local stoquastic: • 2-local on qubits: termwise-stoquastic is stoquastic. • 3-local on qubits: there is a counter-example of a Hamiltonian which is • stoquastic but not term-wise stoquastic (also allowing 1-local unitary • transformations).

14. Quantum k-SAT: remember we have a collection of projectors Pi (m=poly(n) of them). Now we demand that Pi are projectors with nonnegative entries in the computational basis. Thus H=i (I-Pi) is stoquastic. (in fact termwise-stoquastic, i.e. each Hi is stoquastic). How to prove that this problem is a complete problem for the classical class MA? Sketch of ideas: Completeness for a problem Q: 1. Prove that a problem Q is in the class. 2. Prove that any problem in the class can be reduced to Q (“if we can solve Q we can solve any problem in the class”) Stoquastic k-SAT

15. Completeness for a problem Q: 1. Prove that a problem Q is in the class. 2. Prove that any problem in the class can be reduced to Q (“if we can solve Q we can solve any problem in the class”) Kitaev showed 2. for the quantum Ising spin glass by converting Arthur’s quantum verifying circuit into a Hamiltonian. We do 2. by treating a probabilistic verifying circuit as a restricted quantum circuit and applying Kitaev’s construction to obtain a restricted Hamiltonian. This Hamiltonian turns out to be stoquastic! In MA, if answer is yes, verifying circuit with proof always outputs yes (zero error) This results in a Hamiltonian where the ground-state (in the yes-case) satisfies each local constraint, i.e the stoquastic k-SAT problem. Stoquastic k-SAT

16. Containment in MA . Given a set of k-local projectors Pi (m=poly(n) of them) with nonnegative entries in the computational basis and   1/poly(n). -Yes-instance: there is a state that is in the space spanned by all projectors Pi. - No-instance: for any state  there exists an i such that <| Pi |>  1- Consider the nonnegative (symmetric) matrix G= m-1iPi. - Yes-instance: the largest eigenvalue of G is 1. - No-instance: the largest eigenvalue of G is less than 1-/m

17. Containment in MA The idea of the proof: Merlin provides starting point to a random walk on n bit strings. The random walk is a true random walk in case of a Yes-instance. Arthur will test whether the transition probabilities during the walk sum up to 1 and do a final test (which depends on the starting point). In case of a no-instance there is a high probability that some test fails. If a test fails, Arthur decides No.

18. MA (versus NP) Special version of stoquastic k-SAT problem can be viewed as a bunch of constraints (like SAT) and relations. Graph with vertices which are n-bit strings. Vertex x is good when bit-string is in the support of all projectors. We have an edge between vertex x and x’ when there exists an i such that <x|Pi|x’> > 0. Graph has degree m (is poly(n)) Special Stoquastic k-SAT problem that is still MA-complete: Does the graph contain a disconnected sub-graph of only good vertices? How to prove that there is such sub-graph when sub-graph can be exponentially large? Walk around on sub-graph starting from point provided by prover. Promise guarantees that in the no-case after some poly(n) time you will get to a bad vertex. Note that you would explore only a set with poly(n) vertices (chosen by random walk)

19. Discussion Adiabatic evolution with general (local) Hamiltonians is universal for quantum computation. What is the power of adiabatic quantum computation with stoquastic (local) Hamiltonians? Can we classically simulate it? Do heuristic methods/approximation algorithms work better for stoquastic Hamiltonians? Problem is still `hard’ but more ‘classically hard’.