quantum computing reversibility & quantum computing
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Quantum ComputingReversibility &Reversibility &Quantum Computing
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Mandate
Why do all these Quantum Computing guys use reversible logic?
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MaterialLogical reversibility of computationBennett 73 Elementary gates for quantum computationBerenco et al 95 [] quantum computation using teleportationGottesman, Chuang 99
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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
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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
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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
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ExampleInput(11)Workbits
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ExampleInput(11)Workbits
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ExampleInput(11)Workbits
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ExampleInput(11)WorkbitsOutput (1)
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ExampleInput(11)WorkbitsOutputOutput
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ExampleComputeUncomputeCopy Result
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Thermodynamic Reversibilityabcc?b:ac?a:bc
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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
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Quantum State
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Two Distinguishable States
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Continuous State Spaceab+
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Two Spin- Particles
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Four Distinguishable States
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Continuous State Spacec+d+b+a
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Continuous State Space
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Continuous State Space
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State EvolutionH is Hermitian, U is UnitaryLinear, deterministic, reversible(Continuous form)(Discrete form)
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MeasurementOutcome m occurs with probability p(m)Operators Mm non-unitaryProbabilistic, irreversible
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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
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A Simple Measurementab+OutcomeOutcomewith probabilitywith probability
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A Simulated Measurementab+
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A Simulated Measurementab+
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A Simulated Measurementab+
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A Simulated MeasurementTerms remain orthogonal evolve independently, no interferenceor
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Density Operator Representation
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Mixed States
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Partial Trace+AB
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Discarding a Qubit
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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
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Toffoli Gateabc ababc
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Deutschs Controlled-U GateabcabcU
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Equivalent Gate Array=VVVUfor Toffoli gate
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Equivalent Gate ArrayU=CBAP
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Almost Any Gate is Universal
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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
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Protecting against a Bit-Flip (X)EvenOddOddInputOutputSyndrome third qubit flipped(reveals nothing about state)
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Protecting against a Phase-Flip (Z)Phase flip(Z)
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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)
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9-Qubit Shor CodeProtects against all one-qubit errorsError measurements must be erasedImplies heat generation
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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
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Fault Tolerant GatesHS
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Fault Tolerant GatesHHHHHHHencodedcontrolqubit(Steanecode)
encodedtargetqubit(Steanecode)
encodedinputqubit(Steanecode)ZSZSZSZSZSZSZSencodedinputqubit(Steanecode)
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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
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Teleportation CircuitHZM1M2M1XM2
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Simplified Circuit
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Equivalent CircuitHX
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Implementing a GateHXT
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Implementing a GateHSXT
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Implementing a GateHSXT
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ConclusionsQuantum computing requires logical reversibility
Entangled qubits cannot be erased by dispersion
Does not require thermodynamic reversibility
Ancilla preparation, error measurement = refrigerator