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:
σ τ (1) there existsρ an input ρ such that:
σ τ −k F( , ) = Tr σ F() T ( ), 2 I >1−ε
(2) for every inputρ ρ: F() T ( ), 2−k 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 ρinput ρ such that:
F( T ( ), 2−k I )>1−ε
No: for every input ρρ: F( T ( ), 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
ρ
Define quantum transformations T , T as follows: ξ 1 2 ()=Λ()Λ ⊗ ρ † T1 TrM V1 0 0 0 0 V1 ()= † ()⊗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 maxρ F T1 , T2 , 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 ψ ψ ϕ T r = Tr ϕ H 2 H 2
Then there exists a unitary operator U acting only on H 2 such that ψ (I ⊗U ) = 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
ρ ϕ ψ ϕ for any that = 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
ρ ϕ ψ ϕ for any that = ψ purifies T (() ρ ). TrP 1 σ 2 Pr [] accept = max () 1 ⊗ V ϕ σ 2 MaximumMaximum AcceptanceAcceptance ProbabilityProbability (()ρ ) T1
output V qubit V1 V2 M P1 P2 P
ρ ϕ ψ ϕ for any that = ψ purifies T (() ρ ). TrP 1 ϕ 2 Pr [] accept = max V † () 1 ⊗ σ σ 2 MaximumMaximum AcceptanceAcceptance ProbabilityProbability (()ρ ) T1
output V qubit V1 V2 M P1 P2 P
ρ ϕ ψ ϕ for any that = ψ purifies T (() ρ ). TrP 1 ϕ equality forρ best Pr [] accept ≤ max F() T () , 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 maxρ F T1 , T2 , 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. “Honest”“Honest” MerlinMerlin strategystrategy
ρ traced-out 1 Q1 ξ 00Λ 0
ρ traced-out 2 Q2 ξ 00Λ 0 “Honest”“Honest” MerlinMerlin strategystrategy
ψ 1 ϕ 1 Q1 ξ 00Λ 0
ρ traced-out 2 Q2 ξ 00Λ 0 “Honest”“Honest” MerlinMerlin strategystrategy
ψ 1 ϕ 1 Q1 ξ 00Λ 0
ψ 2 ϕ 2 Q2 ξ 00Λ 0 “Honest”“Honest” MerlinMerlin strategystrategy
ψ U 1 ϕ 2 Q1 ξ 00Λ 0 Step 1: ϕ Prepare 2 , send appropriate part to Arthur. ψ 2 ϕ 2 Q2 ξ 00Λ 0 “Honest”“Honest” MerlinMerlin strategystrategy
Step 2 (if b=0) ϕ Send the rest of 2 to Arthur.
Step 1: ϕ Prepare 2 , send appropriate part to Arthur. ψ 2 ϕ 2 Q2 ξ 00Λ 0 “Honest”“Honest” MerlinMerlin strategystrategy
ψ U 1 ϕ 2 Q1 ξ 00Λ 0 Step 1: ϕ Prepare 2 , send appropriate part to Step 2 (if b=1) Arthur. −1 ϕ Apply U to the rest of 2 , rest of to Arthur. QMAMQMAM protocol:protocol: soundnesssoundness
Suppose Merlin is cheating…
Let the reduced state of register R (sent on the first message from Merlin to Arthur) be σ.
Claim: maximumρ acceptanceσ probability is ρ ξ ì ü 1 ()() 2 + 1()σ () 2 max í F T1 , F , Tξ2 ý , î2 2 þ ρ ≤ 1 + 1 ()()ξ () maxρ F T1 , T2 2 2 , ξ OpenOpen questionsquestions
• Find other complete problems for quantum classes.
• Develop relations among complexity of various problems about channels (e.g., problems concerning entanglement, channel capacity,…).
• There are still many interesting questions about quantum interactive proof systems that are unanswered. (E.g., just about everything about QIP(2).)