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Quantum Computing Reversibility & Quantum Computing

Jan 14, 2016

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  • 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