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QMA -complete Problems

QMA -complete Problems. Adam Bookatz December 12, 2012. Quantum-Merlin-Arthur ( QMA ). 1=accept 0=reject . V. s accepts wp , accepts wp. QMA if:. Quantum Circuit SAT ( QCSAT ). Problem: Given a quantum circuit V on n witness qubits determine whether:

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QMA -complete Problems

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  1. QMA-complete Problems Adam Bookatz December 12, 2012

  2. Quantum-Merlin-Arthur (QMA) 1=accept 0=reject V s accepts wp , accepts wp QMA if:

  3. Quantum Circuit SAT (QCSAT) Problem: Given a quantum circuit V on n witness qubits determine whether: (yes case) such that accepts wpb (no case) a promised one of these to be the case where b – a 1/poly(n) • QMA-complete (by definition) 1=accept 0=reject V

  4. Quantum channel property verificationsQMA-complete problems • non-identity check– Given quantum circuit, determine if it is notclose to the identity (up to phase) • non-equivalence check– Given two quantum circuits, determine if they are notapproximately equivalent Given (some type of) quantum channel , determine: • quantum clique – the largest number of inputs states that are still orthogonal after passing through • non-isometry test – whether it does notmap pure states to pure states (even with reference system present) • detecting insecure quantum encryption – whether it is nota private channel • quantum non-expander test – whether it does notsend its input towards the maximally mixed state

  5. Recall… from class that Quantum-k-SATis QMA-complete • We will now look at more general versions • But we require a little bit of physics…

  6. Hamiltonians What is a Hamiltonian, ? • Responsible for time-evolution of a quantum state • Hermitian matrix, • Governs the energy levels of a system • Allowed energy levels are the eigenvalues of • The lowest eigenvalue is called the ground-state energy • Governs the interactions of a system • E.g. simple that acts (nontrivially) only on 2 qubits: • k-local Hamiltonian: where each acts only on kqubits

  7. k-Local Hamiltonian Problem: Given a k-local Hamiltonian on nqubits, , determine whether: (yes case) ground-state energy is a (no case) all of the eigenvalues of are b promised one of these to be the case where b – a 1/poly(n) • QMA-complete for k ≥ 2 (Reduction from QCSAT) • Classical analogue: MAX-k-SAT is NP-complete for k • The k-local terms are like clauses involving k variables • How many of these constraints can be satisfied? (in expectation value) It is in P for k=1

  8. k-Local Hamiltonian There are a plethora of QMA-complete versions: • 2 ≤ k ≤ O(log n) Line with d=11 qudits • geometric locality2-local on 2-D lattice • bosons, fermions • real Hamiltonians • stochastic Hamiltonians (k ≥ 3) • many physically-relevant Hamiltonians • not just ground states: any energy level for • highest energy of a stoquastic Hamiltonian (k ≥ 3 ) Remoevd: bounded strength Hamiltonians (k ≥ 3) Density Functional Theory when only considering eigenvectors that are separable over a partition

  9. Quantum-k-SAT Problem: Given poly(n) many k-local projection operatorson nqubits, determine whether: (yes case) s [cf: k-locHamsaid “ a”] (no case) promised one of these to be the case where 1/poly(n) • k ≥ 4 : QMA1-complete (Reduction from QCSAT) • k = 3 : open question (It is NP-hard) • k = 2 : in P • Classical analogue: k-SAT is NP-complete for k • The k-local terms are like clauses involving kvariables • How many (in expectation value) ofcan these constraints can be satisfied?

  10. Local Consistency of Density Matrices Problem: Given poly(n) many k-local mixed states where each lives only on kqubits of an nqubit space determine whether: (yes case) such that (no case) such that b promised one of these to be the case where b 1/poly(n) • QMA-complete for k ≥ 2 (Reduction from k-local Hamiltonian) • True also for bosonic and fermionic systems

  11. Conclusion • Not so many QMA-complete problems • Contrast: thousands of NP-complete problems • Most important problem is k-local Hamiltonian • Most research has focused on it and its variants • There are a handful of other problems too • Verifying properties of quantum circuits/channels • local consistency of density matrices

  12. QCSAT k-local Hamiltonian [2 ≤ k≤ O(log n)] • constant strength Hamiltonians* • line with 11-state qudits • 2-local on 2-D lattice • interacting bosons, fermions • real Hamiltonians • stochastic Hamiltonians* • physically-relevant Hamiltonians • translationally-invariant Ham. • excited energy level* • highest energy of stoquastic Ham.* • separable k-local Hamiltonian • universal functional of DFT • Channel Property Verification • non-identity check • non-equivalence check • quantum clique • non-isometrytest • detect insecure q. encryption • quantum non-expander test • k-local consistency [k ≥ 2] • bosonic, fermionic • quantum-k-SAT [k≥ 4] • quantum––SAT • quantum––SAT • stochastic-6-SAT * for k ≥ 3

  13. The End

  14. Quantum-k-SAT Problem: Given poly(n) many k-local projection operators on nqubits, determine whether: (yes case) has an eigenvalue of 0 [cf: k-locHamsaid “ a”] (no case) all of the eigenvalues of are b promised one of these to be the case where b 1/poly(n) • Classical analogue: Classical k-SAT is NP-complete for k • The k-local terms are like clauses involving k variables • How many (in expectation value) of these constraints can be satisfied? Equivalently, write it more SAT-like Problem: Given poly(n) many k-local projection operators on nqubits, determine whether: (yes case) such that [exactly] (no case) promised one of these to be the case where 1/poly(n)

  15. Quantum-k-SAT • k ≥ 4 : QMA1-complete (Reduction from: QCSAT) • k = 3 : open question (it is NP-hard) • k= 2 : in P • So the current research focusses around k=3: • quantum––SAT • Same as quantum-k-SAT but • Instead of the qubit being a 2-state qubit it is now a -state qudit QMA1-complete quantum––SAT quantum––SAT

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