From Complexity to Universality
in Quantum Computation
Kae Nemoto
National Institute of Informatics, Tokyo
How Complex is Quantum Computation?
Can a quantum system be probabilistically simulated by a
classical universal computer? … the answer is certainly, No!
— Richard P. Feynman (1982)
N particles of d-level
ϕ1 ⊗Λ ⊗ ϕ N
U
⎯
⎯→
d
∑Λ
j1 =1
d
1
N
c
ϕ
⊗
Λ
⊗
ϕ
∑ j1Κ jN
j N =1
The number of coefficient is in the order of d N
Hilbert space is a big place.
— Carlton Caves
Does the complexity of quantum computation come from Hilbert space?
What makes quantum computation distinct from classical computation?
2
Complexity
•
How can we examine complexity of a quantum system?
The initial state
Hilbert space
Hilbert space
Starting from the initial state at time t, we observe
where the state goes.
Can we say that the right case is more complicated than the left one?
3
Separable and entangled
•
Consider only pure states,
– If the time evolution does not generate any entanglement, then the
state description is rather simple.
ϕ ⊗Λ ⊗ ϕ
1
SU ( d ) ⊗Λ ⊗ SU ( d )
⎯⎯ ⎯ ⎯ ⎯⎯→
N
d
d
N
c
ϕ
⊗
Λ
⊗
c
ϕ
∑ j1
∑ jN
1
j1 =1
j N =1
The number of coefficient is in the order of dN.
– If it is possible to generate entanglement, the state description must
stay in the complicated form:
ϕ ⊗Λ ⊗ ϕ
1
N
SU ( d N )
⎯⎯ ⎯
⎯→
d
∑Λ
j1 =1
d
N
1
c
ϕ
⊗
Λ
⊗
ϕ
∑ j1Κ jN
j N =1
The number of coefficient is in the order of d N.
4
Restricted operations might allow us a simple description
•
•
•
•
Restricting the allowed operations to local unitary operators, we can
have a simple description of states.
The previous example suggests that a system without entanglement
can be represented simply.
Is there any other examples which allows us to have a simple
description of states?
Is it possible to include entangled states in such a simple description?
The schematic image of a simple
state description case
A case which apparently requires
a complex description
The initial state
Hilbert space
Hilbert space
Any way to retrieve the simple description?
If we can do this systematically, then we can evaluate how complex our system is.5
Preparation for the stabilizer formalism
State representation
Operator representation
Qubit case:
Pauli Group
Basis states
0,
Pauli matrices:
1
Find a biggest subgroup which
does not change the state
(including the global phase).
0 : Z0 = 0
(Z 1
⎛0
X = ⎜⎜
⎝1
⎛0
Y = ⎜⎜
⎝i
1⎞
⎟
0 ⎟⎠
−i⎞
⎟
0 ⎟⎠
⎛1 0 ⎞
⎟⎟
Z = ⎜⎜
0
1
−
⎝
⎠
Unity matrix:
⎛1 0⎞
⎟⎟
I = ⎜⎜
0
1
⎝
⎠
⊗ {± 1, i}
= −1 )
Ex. For a given state |0> the subgroup ( I, Z ) does not change the state.
0
↔
(I , Z ) ↔
Z
The subgroup can labeled by the set of generators.
6
Stabilizer Formalism
•
•
•
•
The stabilizer formalism can be a powerful tool to describe quantum
states using group theory.
The group of principal interest is the Pauli group.
Start with 0 in the computational basis.
– A subgroup S = {I , Z } stabilizes the initial state 0 .
– We specify the subgroup by the generator Z .
– We can describe a state via its stabilizer.
Generalized it to multi-qubit systems:
A state:
•
0 ⊗Λ ⊗ 0
The stabilizer generators: Z1 , Z 2 , Κ , Z n
Now, we use the stabilizer formalism to analyze dynamics of systems.
7
Unitary gates and the stabilizer formalism
•
Unitary gates can be considered as a map from one stabilizer to
another.
– U is a unitary gate and the initial state is 0 .
Schrödinger picture
State description
Stabilizer description
0
Z
U
U
U0
UZU +
Heisenberg picture
– We may compute how the unitary gate affects the generators of the
stabilizer.
– The gate U transforms the state <Z > to a linear sum of Pauli group
elements.
8
Special classes of unitary gates
•
In general,
Z
U
UZU +
UZU + = α X + β Y + γ Z , α , β , γ ∈ C
The gate operation transforms the generator Z to an operator
outside of Pauli group.
•
•
•
In general, the stabilizer description is typically as complex as the state
representation.
The final state typically requires 2n amplitudes to be specified.
How can we seek simplicity in quantum computational circuits?
– If we can restrict gate operations to ones which preserve the Pauli
group, then the stabilizer description remains simple.
Z
U
UZU +
U : Hadamard, then UZU + = X
9
Complexity of a gate set
•
•
Using the stabilizer formalism, is there any interesting sets of restricted
gates?
Find a set of gates which preserve the Pauli group.
The initial state can go only
some part of Hilbert space.
Hilbert space
1
…
0
…
n
Cf. Local unitary operations cannot
transform a separable state to a entangled
state.
→ the initial state has to stay
In some part of the Hilbert space.
10
Restrict Dynamics
• Seek the class of operations such that the complexity of the stabilizer
description remains linear in n.
Gate Operations which preserve the Pauli group.
H
X
Z
H=
→
→
S
Z
X
X
Z
1 ⎛1 1 ⎞
⎜⎜
⎟⎟
2 ⎝ 1 − 1⎠
→
→
Y
Z
1
S = Z = ⎛⎜
⎝0
Pauli operators:
→
→
0⎞
i ⎟⎠
⎛1
⎜0
CNOT = ⎜
⎜0
⎝0
Y
X
X
Z
CNOT
X1 → X1 X 2
X2 →
X2
Z1 →
Z1
Z 2 → Z1 Z 2
X
−Z
X
Z
→
→
0
1
0
0
0
0
0
1
0⎞
0⎟
⎟
1⎟
0⎠
Z
−X
−Z
X
Z
→
→
−X
Z
11
One Qubit
S,Z
0
H
X
0 +1
0 +i1
Y
Y
S
S
0 −i 1
X,Z,H
Y,Z
S
S
X
X,Y
0 −1
H
1
S,Z
12
Example: the swap circuit
The swap circuit
+ 1 )⊗ 0
Initial State
0 ⊗( 0 + 1 )
00 + 11
CNOT
(0
CNOT
CNOT
⎧U CNOT ( X 1 ⊗ I 2 )U CNOT +
CNOT operation: ⎨
+
(
)
⊗
U
I
Z
U
CNOT
2
⎩ CNOT 1
X1
X1 X 2
Z2
Z1 Z 2
0 ⊗( 0 + 1 )
=
X1 ⊗ X 2
=
Z1 ⊗ Z 2
X2
Z1
⎧ X 1 ⊗ X 2 ( 00 + 11 )
⎨
⎩ Z1 ⊗ Z 2 ( 00 + 11 )
Final State
X2
Z1
=
=
11 + 00
00 + 11
13
Gottesman-Knill (GK) Theorem
•
GK analyzed: the complexity of quantum computational circuits using
stabilizers.
GK theorem:
A quantum circuit which starts with
a state in the computational basis,
and consists of only
H
Hadamard ⎫
S
Phase ⎪⎪
CNOT Controlled-NOT ⎬⎪
X,Y,Z
Pauli ⎪⎭
gates,
and projective measurements in the computational basis
may be efficiently simulated on a classical computer.
•
GK theorem specifies a class of quantum computations which can be efficiently
simulated on a classical computer.
14
Qubit to Qunat
Qubit
•
Most widely and deeply investigated
computational model.
0,1
Group contraction to Qunats
Easy to translate to each other
Qudit
•
•
0 , 1 , Λ , d −1
Qunat (Continuous Variables)
q and p
Squeezed states
Infinite squeezing
d-level systems
Properties of qudit systems
are quite similar to qubit
systems in many ways.
Squeeze in one direction.
Squeezing Hamiltonian:
(
Hˆ = χ a + 2 + a 2
)
Gaussian states
15
Qunat quantum computation
•
Generalize the Pauli group for qunat quantum computation and seek
the gate operations which preserve the generalized Pauli group.
– The generalized Pauli group:
{
}
The generators qˆ , pˆ , Iˆ ,
[qˆ, pˆ ] = iηδ ij Iˆ
generate the single Pauli operators
X (q) = e
i
− qpˆ
η
, Z ( p) = e
i
pqˆ
η
• X(q) is a position-translation operator (by the amount of q).
• Z(p) is a momentum boost operator (kicking the moment by the
amount of p).
The action of single Pauli operators on the computational basis
of position eigenstates
i
X (q ) s = s + q , Z ( p ) s = exp( sp ) s
η
16
The action of the quibit operators
The qubit operators can be generalized to ones for continuous variables
based on their action:
Hadamard gate H :
X
Z
→
→
Z
X
Phase gate S :
X
Z
→
→
Y
Z
→
→
→
→
X1 ⊗ X 2
I1 ⊗ X 2
Z1 ⊗ I 2
Z1 ⊗ Z 2
CNOT gate :
X1 ⊗ I2
I1 ⊗ X 2
Z1 ⊗ I 2
I1 ⊗ Z 2
17
To preserve the generalized Pauil group
– The special class of unitary operators to preserve the generalized
Pauli group: Fourier transform gate, Phase gate, and SUM gate.
Action
Fourier Transform F :
⎞
⎛i π 2
F = exp⎜
qˆ + pˆ 2 ⎟
⎠
⎝η 4
(
)
Phase gate P :
⎛ i
⎞
P(η ) = exp⎜ η qˆ 2 ⎟
⎝ 2η
⎠
SUM gate:
) ⎞
⎛ i
SUM = exp⎜ − qˆ1 ⊗ p2 ⎟
⎝ η
⎠
X (q )
Z ( p)
X (q )
Z ( p)
→
→
X 1 (q ) ⊗ I 2
Z1 ( p ) ⊗ I 2
I1 ⊗ X 2 (q )
I1 ⊗ Z 2 ( p )
→
→
e
→
→
→
→
Z (q )
X ( p) −1
i
ηq 2
2η
X (q) Z (η q )
Z ( p)
X 1 (q ) ⊗ X 2 (q )
Z1 ( p ) ⊗ I 2
I1 ⊗ X 2 (q )
−1
Z1 ( p ) ⊗ Z 2 ( p )
18
Generalized Gottesman-Knill Theorem
•
The Generalized GK theorem for qunat quantum computation.
– The generalized Pauli group: a set of single-oscillator Pauli
operators.
– The special class of unitary operators to preserve the generalized
Pauli group:
• Fourier transform gate → Hadamard gate
• Phase gate → Phase gate S
• SUM gate → CNOT gate
– Possible measurement: measurements in position- or momentumeigenstate basis with finite losses and classical feed-forward.
•
Physics viewpoint: the simplicity is maintained by quadratic
Hamiltonians which preserve the Gaussian features of the states.
19
Entanglement is not enough
Shall we think such a computation circuit is simple, or rather complicated?
•
•
Simple: A computation circuit which satisfy the GK requirements is
simple enough to efficiently simulate on a classical computer.
Complex: Such a circuit can generate maximally entangled states!
The states which can be generated by the gate set of the GK theorem
does not cover the entire Hilbert space.
Hilbert space
Hilbert space
The initial state
•
Requirements for true quantum computation is much more complicated
than just entanglement.
•
We need other measures to distinguish truly quantum computation
from ones which can be simulated on a classical computer.
20
Beyond GK: Universal sets of gates
It is known for qubit computation that
The gate set of GK theorem
Gates in GK Theorem
+
π/8-gate
Universal gates
Universal sets of gates
efficiently classically simulated
Qudit
SUM gate, etc.
+ arbitrary single-dit transformation
Qubit
CNOT, Hadamard, etc.
+ Single-bit (π/8) rotation
Qunat Quadratic Hamiltonian
•
•
+ third order or beyond
The set of these gates in the GK theorem are not necessarily easy to
realize in a physical system.
Such a classification is nothing to do with physical difficulties in
realization.
21
Gate sets and measurement strategies
•
•
•
•
We have only considered projective measurements in the
computational basis as our measurement strategies.
However if we allow us to use other type of measurement, how may we
change the possible quantum computation?
For instance, Lloyd and Braunstein have shown that ideal photon
counting is enough to complement quadratic Hamiltonian gates to be
universal.
Or we may move to measurement based quantum computation, where
we do not need a universal set of gates to perform universal quantum
computation.
A gate set
Universal set of gates
+
measurements
22
Non-standard quantum computation
Divicenzo’s Criteria
Criterion 1:
An identifiable set of qubits, scalable
in number for large scale quantum
computing.
Criterion 2:
Accurate preparation of initial states.
Criterion 3:
The computation occurs through
control of H and needs conditional
dynamics. (Universal set of gates)
Criterion 4:
The error probability during an
individual gate must be small.
Criterion 5:
Projective quantum measurements
must be possible on chosen qubits.
Operator-based QC
Hamiltonian-based QC
• Adiabatic passage quantum
computation
Gate-based QC
• Quadratic Hamiltonian + Ideal
photon counting
Measurement-based QC
• Cluster-state quantum
computation
Other approaches
• Quantum Cellular automata
• Global addressing
23
Universal quantum computation
Qubit computation
Universal gate sets:
• CNOT + Single-rotation
• CNOT + Hadamard + Phase + π/8-gate
• etc.
This non-universal set can simulate
qubit quantum computation
•
•
Non-universal gates:
Controlled-rotation
etc.
It generates no states with
imaginary components
This set of gates is universal as rebit quantum computation
but not universal in qubit computation.
+ measurements
•
•
CNOT + Hadamard + Phase
etc.
These sets cannot simulate qubit quantum computation.
24
References
•
•
•
Stabilizer formalism:
– M. A. Nielsen and I. L. Chuang, “Quantum Computation and Quantum
Information”, Cambridge University Press, Cambridge (2000).
The GK thorems:
– D. Gottesman, “The Hesenberg Representation of Quantum Computers”,
Group22: Proceedings of the XXII international Colloquium on Group
Theoretical Methods in Physics, eds. S. P. Corney et al, (Cambridge, MA,
International Press, 1999), p. 32, quant-ph/9807006, D. Gottesman, quantph/9802007.
– S. D. Bartlett, B. C. Sanders, S. L. Braunstien, K. Nemoto, Phys. Rev. Lett.,
88, 097904 (2002) .
Universal quantum computation:
– A. Barenco, et. al., Phys. Rev. A, 52, 3457 (1995).
– S. Lloyd, S. L. Braunstein, Phys. Rev. Lett., 82 1784 (1999).
– Y. Shi, quant-ph/0205115.
– E. Farhi, et al., Science 292, 472 (2001).
– M. A. Nielsen, “Optical quantum computation using cluster states”, quantph/0402005.
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