Reversibility & Computing Quantum Computing Reversibility
Transcript of 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
§ Logical reversibility of computationBennett ’73
§ 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
§ Logical reversibility of computationBennett ’73
§ 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
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
§ Logical reversibility of computationBennett ’73
§ 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
Quantum State
↑ ↓
Two Distinguishable States
↑ ↓a b+
Continuous State Space
Two Spin-½ Particles
↑
↓
↑
↑
↓
↓
↓
↑
Four Distinguishable States
c+ d+b+a↑
↓
↑
↑
↓
↓
↓
↑
Continuous State Space
Continuous State Space
Continuous State Space
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 Simulated Measurement
↑ ↓a b+
↑ ↓
A Simulated Measurement
↑ ↓a b+
↑ ↓
A Simulated Measurement
↑ ↓
↑ ↓Terms remain orthogonal –
evolve independently, no interference
or
Density Operator Representation
Mixed States
Partial Trace
+↑
↑
↓
↓
A
B
Discarding a Qubit
Material
§ Logical reversibility of computationBennett ’73
§ 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
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
§ Logical reversibility of computationBennett ’73
§ 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
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
§ Logical reversibility of computationBennett ’73
§ 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
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