QMA-complete Problems

Adam BookatzDecember 12, 2012

Quantum-Merlin-Arthur (QMA)

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¿0… 0 ⟩ V

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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)

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¿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)

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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