Layouts for fault-tolerant quantum computers
Disclosed herein are example layouts and layout generation techniques for fault-tolerant quantum computers. Example embodiments comprise methods for performing a layout reduction technique for fault-tolerant quantum computing. In certain embodiments, a layout of an arbitrary quantum circuit is reduced to a layout of exponents of a multiple qubit Pauli matrix and measurements of a multiple qubit Pauli matrix. In certain embodiments, qubits are marked as one of a data, interface, or ancilla qubit for a 2D nearest neighbor graph of qubit connectivity, and an ancilla-path is provided from a respective data qubit to a respective interface qubit.
1 . A system, comprising:
a quantum computer; and
one or more classical-computing devices, at least some of the one or more classical computing devices being programmed to
transform a quantum computation of the form
C
2
M
(
a
,
Z
i
(
2
)
)
C
1
exp
(
i
ϕZ
i
(
1
)
)
C
0
❘
"\[LeftBracketingBar]"
0
〉
⊗
n
to a sequence of exponentials and measurements of the form:
M
(
a
,
P
2
)
exp
(
i
ϕ
P
1
)
❘
"\[LeftBracketingBar]"
0
〉
⊗
n
in which P 1 and P 2 are multiple qubit Pauli operators,
M(a,P 2 )=(I+aP 2 )/2 denotes measurement of the multiple qubit Pauli operator P 2 with outcome a=±1, wherein/is the identity matrix, wherein
P
1
=
C
0
T
Z
i
(
1
)
)
C
0
,
and
P
2
=
(
C
1
C
0
)
†
M
(
a
,
Z
i
(
2
)
)
C
1
C
0
,
construct a tree (V,E) with root node r∈V, the tree describing a 2D grid of qubits in which each vertex V describes a qubit in the 2D grid, and vertices describing nearest neighbor qubits in the 2D grid are connected by an edge in E; and
configure the quantum computer comprising the 2D grid of qubits to implement the sequence of exponentials and measurements using:
one-qubit measurements and one-qubit Clifford unitaries,
either (A) controlled Z-gates or (B) two-qubit measurements on nearest neighbor qubits described by vertices connected by an edge in E, (A) or (B) being used to perform multi-target controlled Z-gates and each Clifford C and C † such that CZ r C † =P being constructed using multi-target controlled Z-gates, and
single qubit rotations that are only performed on the qubits described by the root node r.
2 . One or more classical-computer-readable-media storing classical-computer-executable instructions, which when executed by a classical computer cause the classical computer to
transform a quantum computation of the form
C
2
M
(
a
,
Z
i
(
2
)
)
C
1
exp
(
i
ϕZ
i
(
1
)
)
C
0
❘
"\[LeftBracketingBar]"
0
〉
⊗
n
to a sequence of exponentials and measurements of the form:
M
(
a
,
P
2
)
exp
(
i
ϕ
P
1
)
❘
"\[LeftBracketingBar]"
0
〉
⊗
n
in which P 1 and P 2 are multiple qubit Pauli operators,
M(a,P 2 )=(I+aP 2 )/2 denotes measurement of the multiple qubit Pauli operator P 2 with outcome a=±1, wherein I is the identity matrix, wherein
P
1
=
C
0
T
Z
i
(
1
)
)
C
0
,
and
P
2
=
(
C
1
C
0
)
†
M
(
a
,
Z
i
(
2
)
)
C
1
C
0
,
construct a tree (V,E) with root node r∈V, the tree describing a 2D grid of qubits in which each vertex V describes a qubit in the 2D grid, and vertices describing nearest neighbor qubits in the 2D grid are connected by an edge in E; and
configure the quantum computer comprising the 2D grid of qubits to implement the sequence of exponentials and measurements using:
one-qubit measurements and one-qubit Clifford unitaries,
either (A) controlled Z-gates or (B) two-qubit measurements on nearest neighbor qubits described by vertices connected by an edge in E, (A) or (B) being used to perform multi-target controlled Z-gates and each Clifford C and C † such that CZ r C † =P being constructed using multi-target controlled Z-gates, and
single qubit rotations that are only performed on the qubits described by the root node r.
3 . A method performed by one or more classical computing devices, comprising:
transforming a quantum computation of the form
C
2
M
(
a
,
Z
i
(
2
)
)
C
1
exp
(
i
ϕZ
i
(
1
)
)
C
0
❘
"\[LeftBracketingBar]"
0
〉
⊗
n
to a sequence of exponentials and measurements of the form:
M
(
a
,
P
2
)
exp
(
i
ϕ
P
1
)
❘
"\[LeftBracketingBar]"
0
〉
⊗
n
in which P 1 and P 2 are multiple qubit Pauli operators, M(a,P 2 )=(I+aP 2 )/2 denotes measurement of the multiple qubit Pauli operator P 2 with outcome a=±1, wherein I is the identity matrix, wherein
P
1
=
C
0
T
Z
i
(
1
)
)
C
0
,
and
P
2
=
(
C
1
C
0
)
†
M
(
a
,
Z
i
(
2
)
)
C
1
C
0
,
constructing a tree (V,E) with root node r∈V, the tree describing a 2D grid of qubits in which each vertex V describes a qubit in the 2D grid, and vertices describing nearest neighbor qubits in the 2D grid are connected by an edge in E; and
configuring a quantum computer comprising the 2D grid of qubits to implement the sequence of exponentials and measurements using:
one-qubit measurements and one-qubit Clifford unitaries,
either (A) controlled Z-gates or (B) two-qubit measurements on nearest neighbor qubits described by vertices connected by an edge in E, (A) or (B) being used to perform multi-target controlled Z-gates and each Clifford C and C † such that CZ r C † =P being constructed using multi-target controlled Z-gates, and
single qubit rotations that are only performed on the qubits described by the root node r.
4 . The method of claim 3 , wherein qubits of the quantum computer comprise data qubits, interface qubits, and ancilla qubits.
5 . The method of claim 4 , where a graph of qubit connectivity satisfies a condition of providing an ancilla-path from one or more data qubits to the interface qubits.
6 . The method of claim 4 , wherein every data qubit is connected to an interface qubit via a path of ancilla qubits.
7 . A system, comprising:
a quantum computer; and
one or more classical-computing devices, at least some of the one or more classical computing devices being programmed to:
transform a quantum computation of the form
C
2
M
(
a
,
Z
i
(
2
)
)
C
1
exp
(
i
ϕZ
i
(
1
)
)
C
0
❘
"\[LeftBracketingBar]"
0
〉
⊗
n
to a sequence of exponentials and measurements of the form:
M
(
a
,
P
2
)
exp
(
i
ϕ
P
1
)
❘
"\[LeftBracketingBar]"
0
〉
⊗
n
in which P 1 and P 2 are multiple qubit Pauli operators,
M(a,P 2 )=(1+aP 2 )/2 denotes measurement of the multiple qubit Pauli operator P 2 with outcome a=±1, wherein I is the identity matrix, wherein
P
1
=
C
0
T
Z
i
(
1
)
)
C
0
,
and
P
2
=
(
C
1
C
0
)
†
M
(
a
,
Z
i
(
2
)
)
C
1
C
0
,
construct a tree (V,E) with root node r∈V, the tree describing a 2D grid of qubits in which each vertex V describes a qubit in the 2D grid, and vertices describing nearest neighbor qubits in the 2D grid are connected by an edge in E; and
configure the quantum computer comprising the 2D grid of qubits to implement the sequence of exponentials and measurements using:
one-qubit measurements and one-qubit Clifford unitaries,
either (A) controlled Z-gates or (B) two-qubit measurements on nearest neighbor qubits described by vertices connected by an edge in E, (A) or (B) being used to perform multi-target controlled Z-gates and each Clifford C and C † such that CZ r C † =P being constructed using multi-target controlled Z-gates, and
single qubit rotations that are only performed on the qubits described by the root node r, wherein qubits of the quantum computer comprise data qubits, interface qubits, and ancilla qubits and a graph of qubit connectivity satisfies a condition of providing an ancilla-path from one or more data qubits to the interface qubits.