Reversibility & Computing Quantum Computing Reversibility

Quantum ComputingReversibility &

Reversibility &

Quantum Computing

Mandate

“Why do all these Quantum Computing guys use reversible logic?”

Material

§ Logical reversibility of computationBennett ’73

§ Elementary gates for quantum computationBerenco et al ’95

§ […] quantum computation using teleportationGottesman, Chuang ’99

Material


§ Quantum computing needs logical reversibility§ Elementary gates for quantum computation

Berenco et al ’95§ Gates can be thermodynamically irreversible§ […] quantum computation using teleportation

Gottesman, Chuang ’99

Material





Heat Generation in Computing

§ Landauer’s Principle– Want to erase a random bit? It will cost you– Storing unwanted bits just delays the inevitable

§ Bennett’s Loophole– Computed bits are not random– Can uncompute them if we’re careful

Example

Input(11)

Workbits

Example

Input(11)

Workbits

Example

Input(11)

Workbits

Example

Input(11)

Workbits

Output (1)

Example

Input(11)

Workbits

OutputOutput

Example

Compute UncomputeCopy Result

Thermodynamic Reversibility

a

bc

c?b:a

c?a:bc

Material





Quantum State

↑ ↓

Two Distinguishable States

↑ ↓a b+

Continuous State Space

Two Spin-½ Particles

↑

↓

↑

↑

↓

↓

↓

↑

Four Distinguishable States

c+ d+b+a↑

↓

↑

↑

↓

↓

↓

↑




State Evolution

§ H is Hermitian, U is Unitary§ Linear, deterministic, reversible

h (Continuous form)

(Discrete form)

Measurement

§ Outcome m occurs with probability p(m)§ Operators Mm non-unitary§ Probabilistic, irreversible

Deriving Measurement

“It can be done up to a point… But it becomes embarrassing to the spectators even before it becomes uncomfortable for the snake”

– Bell

“Like a snake trying to

swallow itself by the tail”

A Simple Measurement

↑ ↓a b+

↑

↓

Outcome

Outcome with probability

with probability

A Simulated Measurement

↑ ↓a b+

↑


↑ ↓a b+

↑ ↓


↑ ↓a b+

↑ ↓


↑ ↓

↑ ↓Terms remain orthogonal –

evolve independently, no interference

or

Density Operator Representation

Mixed States

Partial Trace

+↑

↑

↓

↓

A

B

Discarding a Qubit

Material





Toffoli Gate

a

b

c ⊕ ab

a

b

c

Deutsch’s Controlled-U Gate

a

b

c’

a

b

c U

for Toffoli gate

=

V V† VU

Equivalent Gate Array

for Toffoli gate

U=

C B A

P

Equivalent Gate Array

Almost Any Gate is Universal

Material





Protecting against a Bit-Flip (X)

Even Odd OddInput Output

Syndrome à third qubit flipped(reveals nothing about state)

Protecting against a Phase-Flip (Z)

Phase flip(Z)

General Errors

§ Pauli matrices form basis for 1-qubit operators:

§ I is identity, X is bit-flip, Z is phase-flip§ Y is bit-flip and phase-flip combined (Y = iXZ)

9-Qubit Shor Code

§ Protects against all one-qubit errors§ Error measurements must be erased§ Implies heat generation

Material





Fault Tolerant Gates

H

S

Fault Tolerant GatesHHHHHHH

encodedcontrolqubit(Steanecode)

encodedtargetqubit(Steanecode)

encodedinputqubit(Steanecode)

ZSZSZSZSZSZSZS

encodedinputqubit(Steanecode)

Clifford Group

§ Encoded operators are tricky to design

§ Manageable for operators in Clifford group using stabilizer codes, Heisenberg representation

§ Map Pauli operators to Pauli operators

§ Not universal

H

ZM1

M2

M1

Teleportation Circuit

XM2

Simplified Circuit

H X

Equivalent Circuit

H X T

Implementing a Gate

H SXT

Implementing a Gate

H SXT

Implementing a Gate

Conclusions

§ Quantum computing requires logical reversibility

– Entangled qubits cannot be erased by dispersion

§ Does not require thermodynamic reversibility

– Ancilla preparation, error measurement = refrigerator

Documents

Reversibility & Computing Quantum Computing Reversibility