Post on 24-Dec-2015
QMA-complete Problems
Adam BookatzDecember 12, 2012
Quantum-Merlin-Arthur (QMA)
¿𝜓 ⟩
¿0… 0 ⟩ V
¿𝜓 ⟩
s accepts wp
, accepts wp
1=accept0=reject
QMA if:
QUANTUM CIRCUIT SAT (QCSAT)
• QMA-complete (by definition)
Problem: Given a quantum circuit V on n witness qubitsdetermine whether:
(yes case) such that accepts wp b(no case) a
promised one of these to be the casewhere b – a 1/poly(n)
¿𝜓 ⟩
¿0… 0 ⟩ V1=accept0=reject
QUANTUM CHANNEL PROPERTY VERIFICATIONSQMA-COMPLETE problems
• NON-IDENTITY CHECK – Given quantum circuit, determine if it is not close to the identity (up to phase)
• NON-EQUIVALENCE CHECK – Given two quantum circuits, determine if they are not approximately 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 not map pure states to pure states (even with reference system present)
• DETECTING INSECURE QUANTUM ENCRYPTION – whether it is not a private channel
• QUANTUM NON-EXPANDER TEST – whether it does not send its input towards the maximally mixed state
Recall…from class that QUANTUM-k-SAT is QMA-complete
• We will now look at more general versions
• But we require a little bit of physics…
HamiltoniansWhat 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 k qubits
k-LOCAL HAMILTONIAN
• 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)
Problem: Given a k-local Hamiltonian on n qubits, ,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 casewhere b – a 1/poly(n)
It is in P for k=1
• 2 ≤ k ≤ O(log n)
Line with d=11 qudits• geometric locality 2-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 )
k-LOCAL HAMILTONIANThere are a plethora of QMA-complete versions:
QUANTUM-k-SAT
• 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 k variables– How many (in expectation value) of can these constraints can be
satisfied?
Problem: Given poly(n) many k-local projection operators on n qubits, determine whether:
(yes case) s [cf: k-LOCHAM said “ a”](no case)
promised one of these to be the casewhere 1/poly(n)
LOCAL CONSISTENCY OF DENSITY MATRICES
• QMA-complete for k ≥ 2 (Reduction from k-LOCAL HAMILTONIAN)
• True also for bosonic and fermionic systems
Problem: Given poly(n) many k-local mixed states where each lives only on k qubits of an n qubit spacedetermine whether:
(yes case) such that (no case) such that b
promised one of these to be the casewhere b 1/poly(n)
𝝆𝟏
𝝆𝟑𝝆𝟐
𝝈
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
CHANNEL PROPERTY VERIFICATION• NON-IDENTITY CHECK • NON-EQUIVALENCE CHECK• QUANTUM CLIQUE• NON-ISOMETRY TEST• DETECT INSECURE Q.
ENCRYPTION• QUANTUM NON-EXPANDER
TEST
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 DFTk-LOCAL CONSISTENCY [k ≥ 2]
• bosonic, fermionic
QCSAT
QUANTUM-k-SAT [k ≥ 4] • QUANTUM––SAT• QUANTUM––SAT• STOCHASTIC-6-SAT
* for k ≥ 3
The End
QUANTUM-k-SAT
• 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?
Problem: Given poly(n) many k-local projection operators on n qubits, determine whether:
(yes case) has an eigenvalue of 0 [cf: k-LOCHAM said “ a”](no case) all of the eigenvalues of are b
promised one of these to be the casewhere b 1/poly(n)
Problem: Given poly(n) many k-local projection operators on n qubits, determine whether:
(yes case) such that [exactly](no case)
promised one of these to be the casewhere 1/poly(n)
Equivalently, write it more SAT-like
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 quditQUANTUM––SATQUANTUM––SAT QMA1-complete