Quantum ComputingReversibility &Reversibility &Quantum Computing
Mandate
Why do all these Quantum Computing guys use reversible logic?
MaterialLogical reversibility of computationBennett 73 Elementary gates for quantum computationBerenco et al 95 [] quantum computation using teleportationGottesman, Chuang 99
MaterialLogical reversibility of computationBennett 73Quantum computing needs logical reversibilityElementary gates for quantum computationBerenco et al 95Gates can be thermodynamically irreversible[] quantum computation using teleportationGottesman, Chuang 99
MaterialLogical reversibility of computationBennett 73Quantum computing needs logical reversibilityElementary gates for quantum computationBerenco et al 95Gates can be thermodynamically irreversible[] quantum computation using teleportationGottesman, Chuang 99
Heat Generation in ComputingLandauers PrincipleWant to erase a random bit? It will cost youStoring unwanted bits just delays the inevitable
Bennetts LoopholeComputed bits are not randomCan uncompute them if were careful
ExampleInput(11)Workbits
ExampleInput(11)Workbits
ExampleInput(11)Workbits
ExampleInput(11)WorkbitsOutput (1)
ExampleInput(11)WorkbitsOutputOutput
ExampleComputeUncomputeCopy Result
Thermodynamic Reversibilityabcc?b:ac?a:bc
MaterialLogical reversibility of computationBennett 73Quantum computing needs logical reversibilityElementary gates for quantum computationBerenco et al 95Gates can be thermodynamically irreversible[] quantum computation using teleportationGottesman, Chuang 99
Quantum State
Two Distinguishable States
Continuous State Spaceab+
Two Spin- Particles
Four Distinguishable States
Continuous State Spacec+d+b+a
Continuous State Space
Continuous State Space
State EvolutionH is Hermitian, U is UnitaryLinear, deterministic, reversible(Continuous form)(Discrete form)
MeasurementOutcome m occurs with probability p(m)Operators Mm non-unitaryProbabilistic, irreversible
Deriving MeasurementIt can be done up to a point But it becomes embarrassing to the spectators even before it becomes uncomfortable for the snake BellLike a snake trying to swallow itself by the tail
A Simple Measurementab+OutcomeOutcomewith probabilitywith probability
A Simulated Measurementab+
A Simulated Measurementab+
A Simulated Measurementab+
A Simulated MeasurementTerms remain orthogonal evolve independently, no interferenceor
Density Operator Representation
Mixed States
Partial Trace+AB
Discarding a Qubit
MaterialLogical reversibility of computationBennett 73Quantum computing needs logical reversibilityElementary gates for quantum computationBerenco et al 95Gates can be thermodynamically irreversible[] quantum computation using teleportationGottesman, Chuang 99
Toffoli Gateabc ababc
Deutschs Controlled-U GateabcabcU
Equivalent Gate Array=VVVUfor Toffoli gate
Equivalent Gate ArrayU=CBAP
Almost Any Gate is Universal
MaterialLogical reversibility of computationBennett 73Quantum computing needs logical reversibilityElementary gates for quantum computationBerenco et al 95Gates can be thermodynamically irreversible[] quantum computation using teleportationGottesman, Chuang 99
Protecting against a Bit-Flip (X)EvenOddOddInputOutputSyndrome third qubit flipped(reveals nothing about state)
Protecting against a Phase-Flip (Z)Phase flip(Z)
General ErrorsPauli matrices form basis for 1-qubit operators:
I is identity, X is bit-flip, Z is phase-flipY is bit-flip and phase-flip combined (Y = iXZ)
9-Qubit Shor CodeProtects against all one-qubit errorsError measurements must be erasedImplies heat generation
MaterialLogical reversibility of computationBennett 73Quantum computing needs logical reversibilityElementary gates for quantum computationBerenco et al 95Gates can be thermodynamically irreversible[] quantum computation using teleportationGottesman, Chuang 99
Fault Tolerant GatesHS
Fault Tolerant GatesHHHHHHHencodedcontrolqubit(Steanecode)
encodedtargetqubit(Steanecode)
encodedinputqubit(Steanecode)ZSZSZSZSZSZSZSencodedinputqubit(Steanecode)
Clifford GroupEncoded operators are tricky to design
Manageable for operators in Clifford group using stabilizer codes, Heisenberg representation
Map Pauli operators to Pauli operators
Not universal
Teleportation CircuitHZM1M2M1XM2
Simplified Circuit
Equivalent CircuitHX
Implementing a GateHXT
Implementing a GateHSXT
Implementing a GateHSXT
ConclusionsQuantum computing requires logical reversibility
Entangled qubits cannot be erased by dispersion
Does not require thermodynamic reversibility
Ancilla preparation, error measurement = refrigerator
