CapturingCapturing quantumquantum complexitycomplexity classesclasses viavia quantumquantum channelschannels John Watrous Department of Computer Science University of Calgary QuantumQuantum channelschannels In this talk, a quantum channel is just a trace-preserving, completely positive mapping from n qubits to k qubits. Described by (unitary) quantum circuits: ρ traced-out Q ξ 00Λ 0 T ()ρ = ξ GeneralGeneral typetype ofof problemproblem We’ll be interested in the following general type of computational problem: Input: classical description of one or more quantum channels Promise: some guarantee on the properties of the channel or channels. Output: “yes” or “no” ExampleExample problemproblem #1:#1: cancan thethe outputoutput bebe closeclose toto totallytotally mixed?mixed? Input: a quantum channel T. Promise: one of the following holds: (1) there exists an input ρ such that: −k F(σ ,τ ) = Tr σ τ σ F() T (ρ), 2 I >1−ε (2) for every input ρ: −k F() T (ρ), 2 I < ε Output: “yes” if (1) holds, “no” if (2) holds. ExampleExample problemproblem #1:#1: cancan thethe outputoutput bebe closeclose toto totallytotally mixed?mixed? Shorthand for same problem: Input: a quantum channel T. Yes: there exists an input ρ such that: F( T (ρ), 2−k I )>1−ε No: for every input ρ: F( T (ρ), 2−k I )< ε ExampleExample problemproblem #2:#2: outputsoutputs closeclose together?together? Input: two quantum channels T 1 and T 2 . Yes: there exist states ρ and ξ such that: ()ρ ()ξ > −ε F T1( ), T2 1 No: for all states ρ and ξ: ()ρ ()ξ < ε F T1( ), T2 SpecialSpecial case:case: n=n=00 It makes sense to consider “channels” with no input: traced-out 00Λ 0 Q ξ We won’t refer to these as channels… Convention: when we want to describe states, we will describe them in this way. ExampleExample problemproblem #3:#3: outputoutput closeclose toto aa givengiven state?state? Input: a quantum channel T and a state ξ. Yes: there exist a state ρ such that: F() T (ρ), ξ >1−ε No: for all states ρ: F() T (ρ), ξ < ε ExampleExample problemproblem #4:#4: statesstates closeclose together?together? Input: quantum states ρ and ξ. Yes: ρ and ξ are close together: F() ρ, ξ >1−ε No: ρ and ξ are far apart: F() ρ, ξ < ε ExampleExample problemproblem #5:#5: entanglemententanglement breakingbreaking channel?channel? Input: a quantum channel T. Yes: T is close to entanglement breaking: for all ρ there exists separable ξ s.t. F() (T ⊗ I)(ρ), ξ >1−ε No: T is far from entanglement breaking: there exists ρ s.t. for all separable ξ: F() (T ⊗ I)(ρ), ξ < ε CompleteComplete problemsproblems Let A denote some promise problem. Write Ayes , Ano to denote sets of “yes” instances and “no” instances, respectively. Promise problem A is complete for class C if: 1. A∈C 2. for all promise problems B in C there exists a polynomial-time computable f such that ∈ Þ ∈ ∈ Þ ∈ x Byes f (x) Ayes, x Bno f (x) Ano SimpleSimple exampleexample Consider the following problem A: Input: a quantum state ρ on one qubit. … ng Yes: 1 ρ 1 ≥ 2/3 ti es er ρ ≤ nt No: 1 1 1/3 y i er t v 1. Easily solved no in BQP (by simulating the circuit is that describess ρ). Thi 2. Any promise problem B in BQP reduces to A (by virtue of the fact that there exists an efficient quantum algorithm for B). QuantumQuantum InteractiveInteractive ProofProof SystemsSystems x x Prover Verifier quantum channel quantum, quantum no polynomial computational time restriction accept if the verifier believes x is a “yes” input Output: reject otherwise ProblemsProblems withwith quantumquantum interactiveinteractive proofsproofs A promise problem A has a quantum interactive proof system if there exists a verifier V such that: 1. (completeness condition) ∈ If x A yes then there exists some prover P that convinces V to accept (with high probability). 2. (soundness condition) ∈ If x A no then no prover P can convince V to accept (except with small probability). ExampleExample problemproblem #2:#2: outputsoutputs closeclose together?together? Input: two quantum channels T 1 and T 2 . Yes: there exist states ρ and ξ such that: ()ρ ()ξ > −ε F T1( ), T2 1 No: for all states ρ and ξ: ()ρ ()ξ < ε F T1( ), T2 Complete for QIP. ExampleExample problemproblem #3:#3: outputoutput closeclose toto aa givengiven state?state? Input: a quantum channel T and a state ξ. Yes: there exist a state ρ such that: F() T (ρ), ξ >1−ε No: for all states ρ: F() T (ρ), ξ < ε Complete* for QIP(2). ExampleExample problemproblem #4:#4: statesstates closeclose together?together? Input: quantum states ρ and ξ. Yes: ρ and ξ are close together: F() ρ, ξ >1−ε No: ρ and ξ are far apart: F() ρ, ξ < ε Complete for QSZK HV . BackBack toto exampleexample problemproblem #2:#2: outputsoutputs closeclose together?together? Input: two quantum channels T 1 and T 2 . Yes: there exist states ρ and ξ such that: ()ρ ()ξ > −ε F T1( ), T2 1 No: for all states ρ and ξ: ()ρ ()ξ < ε F T1( ), T2 Complete for QIP. 33--MessageMessage QuantumQuantum InteractiveInteractive ProofsProofs We know that QIP = QIP(3), so we just need to show that any problem B with a 3-message quantum interactive proof reduces to our problem. output verifier’s qubit qubits V1 V2 message qubits P1 P2 prover’s qubits message 1 message 2 message 3 33--MessageMessage QuantumQuantum InteractiveInteractive ProofsProofs We know that QIP = QIP(3), so we just need to show that any problem B with a 3-message quantum interactive proof reduces to our problem. V V1 V2 M P1 P2 P message 1 message 2 message 3 RemovingRemoving proverprover fromfrom thethe picturepicture output verifier’s qubit qubits V1 V2 message qubits P1 P2 prover’s qubits TransformationsTransformations T1 T2 1 00Λ 0 V1 V2 Define quantum transformations T 1 , T 2 as follows: ()ρ =Λ()Λ ⊗ ρ † T1 TrM V1 0 0 0 0 V1 ()ξ = † ()⊗ξ T2 TrM V2 1 1 V2 MaximumMaximum AcceptanceAcceptance ProbabilityProbability The maximum probability with which the prover can convince the verifier to accept is: ()()ρ ()ξ 2 max F T1 , T2 ρ , ξ where the maximum is over all inputs ρ and ξ. BipartiteBipartite QuantumQuantum StatesStates Suppose ψ and ϕ are bipartite quantum states ψ ϕ ∈ ⊗ , Η 1 H 2 that satisfy T r ψ ψ = Tr ϕ ϕ H 2 H 2 Then there exists a unitary operator U acting only on H 2 such that (I ⊗U ) ψ = ϕ (Approximate version also holds.) OptionsOptions forfor thethe ProverProver ρ verifier’s qubits V V message qubits P prover’s qubits ψ ϕ Prover can transform ψ to ϕ for any ϕ that leaves the verifier’s qubits in state ρ . MaximumMaximum AcceptanceAcceptance ProbabilityProbability output V qubit V1 V2 M P1 P2 P ψ MaximumMaximum AcceptanceAcceptance ProbabilityProbability (()ρ) T1 output V qubit V1 V2 M P1 P2 P ϕ ϕ for any that ρ = ψ ψ purifies T (() ρ ) . TrP 1 [[]] = Π ϕ 2 Pr accept acc V2 MaximumMaximum AcceptanceAcceptance ProbabilityProbability (()ρ) T1 output V qubit V1 V2 M P1 P2 P ϕ ϕ for any that ρ = ψ ψ purifies T (() ρ ) . TrP 1 2 Pr [] accept = max () 1 ⊗ σ V ϕ σ 2 MaximumMaximum AcceptanceAcceptance ProbabilityProbability (()ρ) T1 output V qubit V1 V2 M P1 P2 P ϕ ϕ for any that ρ = ψ ψ purifies T (() ρ ) . TrP 1 2 Pr [] accept = max ϕ V † () 1 ⊗ σ σ 2 MaximumMaximum AcceptanceAcceptance ProbabilityProbability (()ρ) T1 output V qubit V1 V2 M P1 P2 P ϕ ϕ for any that ρ = ψ ψ purifies T (() ρ ) . TrP 1 equality for best ϕ Pr [] accept ≤ max F() T ()ρ , T ()ξ 2 ξ 1 2 MaximumMaximum AcceptanceAcceptance ProbabilityProbability The maximum probability with which the prover can convince the verifier to accept is: ()()ρ ()ξ 2 max F T1 , T2 ρ , ξ where the maximum is over all inputs ρ and ξ. ApplicationsApplications The completeness of these problems allows us to prove various things about the corresponding classes, such as: •QIP ⊆ EXP : the complete problem can be solved in EXP via semidefinite programming. • QSZK closed under complement and parallelizable to 2 messages: there exists a 2-message QSZK-protocol for the corresponding complete problem and its complement •QSZK ⊆ PSPACE : the complete problem can be solved in PSPACE. ApplicationsApplications •QIP ⊆ QMAM : any problem having a quantum interactive proof system also has a 3-message quantum Arthur-Merlin game. A quantum Arthur-Merlin game is a restricted type of quantum interactive proof: • the verifier’s (Arthur’s) messages consist only of fair coin-flips (classical). • all of Arthur’s computation takes place after all messages have been sent. QuantumQuantum ArthurArthur--MerlinMerlin protocolprotocol ρ traced-out 1 Q1 ξ S 00Λ 0 ancilla R ρ traced-out 2 Q2 ξ 00Λ 0 QuantumQuantum ArthurArthur--MerlinMerlin protocolprotocol Message 1 (from Merlin to Arthur): Merlin sends some register R (supposedly corresponding to the common output of the channels ). Message 2 (from Arthur to Merlin): Arthur flips a coin: call the result b. Send b to Merlin. Message 3 (from Merlin to Arthur): Merlin sends some register S (corresponds to traced-out qubits ). QuantumQuantum ArthurArthur--MerlinMerlin protocolprotocol Arthur’s verification procedure (after messages are sent): If the coin-flip was b = 0: -1 Apply Q 2 to (R,S). Accept if all ancilla qubits are set to 0, reject otherwise. If the coin-flip was b = 1: -1 Apply Q 1 to (R,S). Accept if all ancilla qubits are set to 0, reject otherwise.