Shift rule for gradient determination in parameterised quantum evolutions
A hybrid computing system configured to estimate a derivative of a parameter-dependent physical quantity that is dependent on a first control parameter θ of a parameterised quantum evolution executed by the quantum computing system. The classical computing system determines, based on an input bounding value and phase-correction value, a subset multiset of shift-values which define an equivalent subset multiset of trial control parameter values and corresponding weighting values and summation factors such that the trial control parameter values obey a targeted probability distribution. The quantum computing system executes the parameterised quantum evolution at the trial control parameter values and measures a parameter-dependent physical quantity for each trial control parameter value. On the classical computing system, the measurement results from these measurements are combined with the weighting values and summation factors to calculate at an estimate of a phase-correction-scaled derivative of the parameter-dependent physical quantity.
1 . A method for performing a parameterised quantum evolution for a quantum system associated with a hybrid computing system having a quantum computing system and a classical computing system, wherein the quantum computing system comprises a quantum system having one or more quantum devices,
the method comprising estimating a derivative of a parameter-dependent physical quantity that is dependent on a first control parameter θ of the parameterised quantum evolution by:
on the classical computing system:
determining a finite sub-multi-set S from the set of shift values
{
(
n
-
1
2
)
2
K
❘
n
∈
ℤ
}
,
wherein K is a bounding value defining the interval [φ−K, φ+K] which contains the Fourier spectrum of an expectation-value function ƒ(θ) of the first parameterised unitary quantum evolution, where φ is a phase-correction value;
determining a finite sub-multi-set of weighting values q s each weighting value corresponding to a shift value s from the sub-multi-set of shift values S;
wherein the sub-multi-set of shift values and the respective sub-multi-set of weighting values are determined in accordance with a targeted probability distribution
P
(
s
)
=
1
(
2
π
K
s
)
2
;
calculating a sub-multi-set of trial control parameter values, each trial control parameter value generated by combining a respective shift value s from the sub-multi-set of shift values S with the first value of the first control parameter θ 0 ;
calculating a sub-multi-set of summation factors d s , each summation factor d s =2πK(−1) 2Ks+1/2 for a respective shift value s from the sub-multi-set of shift values S;
for each trial control parameter value:
communicating the trial control parameter value from the classical computing system to the quantum computing system; and
on the quantum computing system:
subjecting the quantum system to a parameterised unitary quantum evolution with the first control parameter of the parameterised unitary quantum evolution set to the trial control parameter value;
making a quantum measurement on the quantum state to obtain a parameter-dependent physical quantity F s ;
communicating the parameter-dependent physical quantity to the classical computing system; and
on the classical computing system:
calculating a phase-corrected measurement result F sφ for each shift value s by calculating F s e −2πi(θ 0 −s)φ ; and
calculating the estimate of the derivative of ƒ φ (θ)=e −2πi(θ)φ ƒ(θ) with respect to the first control parameter through the summation
f
φ
′
(
θ
0
)
=
∑
s
∈
S
d
s
q
s
F
s
φ
.
2 . The method of claim 1 , wherein the phase-correction value is zero such that the interval which contains the Fourier spectrum of the expectation-value function of the first parameterised unitary quantum evolution is defined by [−K, +K], the phase-corrected measurement result F s e −2π(θ 0 −s)φ =F s , and the estimate of the derivative calculated by the classical computing system is
f
′
(
θ
0
)
=
∑
s
∈
S
d
s
q
s
F
s
.
3 . The method of claim 1 , wherein the method further comprises calculating the estimate of the derivative of the expectation-value function ƒ′(θ 0 ) from ƒ φ ′(θ 0 ).
4 . The method of claim 1 , wherein generating the sub-multi-set of shift values and the sub-multi-set of weighting values according to a targeted probability distribution involves:
until a target size N of the sub-multi-set of shift values is reached: randomly sampling a first sampled value s from
{
(
n
-
1
2
)
2
K
❘
n
∈
ℤ
}
according to the targeted probability distribution
P
(
s
)
=
1
(
2
π
K
s
)
2
,
and adding the first sampled value s to the sub-multi-set of shift values S; and
setting all weighting values q s to unity.
5 . The method of claim 1 , wherein generating the sub-multi-set of shift values and the sub-multi-set of weighting values according to a targeted probability distribution involves:
until a target size N of the sub-multi-set of shift values is reached: randomly sampling a first sampled value s from
{
(
n
-
1
2
)
2
K
❘
n
∈
ℤ
,
n
≥
1
}
,
according to the targeted probability distribution
P
(
s
)
=
2
(
2
π
K
s
)
2
,
and adding the first sampled value s and the additive inverse of the sampled value −s to the sub-multi-set of shift values; and
setting all weighting values q s , q −s to unity.
6 . The method of claim 1 , wherein generating the sub-multi-set of shift values and the sub-multi-set of weighting values according to a targeted probability distribution involves:
selecting a target size N,
constructing a set of values
S
=
{
(
n
-
1
2
)
2
K
❘
n
∈
ℤ
}
for n=−N+1, . . . , N;
and determining weighting values
q
s
=
1
(
2
π
K
s
)
2
.
7 . The method of claim 1 , wherein generating the sub-multi-set of shift values and the sub-multi-set of weighting values according to a targeted probability distribution involves:
selecting a target size N,
selecting a positive integer T much smaller than half the target size,
constructing a set of values
S
=
{
(
n
-
1
2
)
2
K
❘
n
∈
ℤ
,
n
≥
1
}
⋂
[
-
(
T
-
1
/
2
)
/
2
K
,
+
(
T
-
1
/
2
)
/
2
K
]
by taking s N s times in such a way as to minimise the variance of the random variable
∑
s
∈
{
(
n
-
1
2
)
/
2
K
❘
n
∈
ℤ
,
n
≥
1
}
⋂
[
-
(
T
-
1
/
2
)
/
2
K
,
+
(
T
-
1
/
2
)
/
2
K
]
(
-
1
)
2
K
s
+
1
2
π
(
2
K
s
)
2
Avg
N
s
(
F
s
)
.
8 . The method of claim 1 , wherein calculating a sub-multi-set of trial control parameter values further comprises passing each trial control parameter value through a parameter-folding function, to modify the value of the trial control parameter value such that no trial control parameter value differs from the first control parameter value by more than a parameter shift limit value.
9 . The method of claim 8 , wherein parameter-folding function has the property τ(x)−x∈p ={pk|k ∈ }, where
p
:=
k
γ
,
where ρ is a real number, k is a positive integer and γ is a is a common divisor of the eigenvalues of the first parameterised quantum evolution.
10 . The method of claim 9 , wherein the parameter-folding function is
τ
(
x
)
=
{
-
c
-
(
(
-
x
)
%
p
)
,
if
x
≤
-
c
-
p
;
x
,
if
x
∈
]
-
c
-
p
,
+
c
+
p
[
;
+
c
+
(
x
%
p
)
,
if
x
≥
+
c
+
p
;
where c is a real number, c=m·p where m is a positive integer, and where % p stands for the remainder in [0, p[ when dividing by p.
11 . The method of claim 1 , wherein the bounding value K is based on the largest and smallest eigenvalue of the first parameterised quantum evolution.
12 . A hybrid computing system comprising a classical computing system and a quantum computing system, the quantum computing system comprising a quantum system having one or more quantum devices, the hybrid computing system configured to:
on the classical computing system:
determine a finite sub-multi-set S from the set of shift values
{
(
n
-
1
2
)
2
K
❘
n
∈
ℤ
}
,
wherein K is a bounding value defining the interval [φ−K, φ+K] which contains the Fourier spectrum of the expectation-value function ƒ(θ) of the first parameterised unitary quantum evolution, where φ is a phase-correction value;
determine a finite sub-multi-set of weighting values q s each weighting value corresponding to a shift value s from the sub-multi-set of shift values S;
wherein the sub-multi-set of shift values and the respective sub-multi-set of weighting values are determined in accordance with a targeted probability distribution
P
(
s
)
=
1
(
2
π
Ks
)
2
;
calculate a sub-multi-set of trial control parameter values, each trial control parameter value generated by combining a respective shift value s from the sub-multi-set of shift values S with the first value of the first control parameter θ 0 ;
calculate a sub-multi-set of summation factors d s , each summation factor d s =2πK(−1) 2Ks+1/2 for a respective shift value s from the sub-multi-set of shift values S;
for each trial control parameter value:
communicate the trial control parameter value from the classical computing system to the quantum computing system; and
on the quantum computing system:
subject the quantum system to a parameterised unitary quantum evolution with the first control parameter of the parameterised unitary quantum evolution set to the trial control parameter value;
make a quantum measurement on the quantum state to obtain a parameter-dependent physical quantity F s ;
communicate the parameter-dependent physical quantity to the classical computing system; and
on the classical computing system:
calculate a phase-corrected measurement result F sφ for each shift value s by calculating F s e −2πi(θ 0 −s)φ ; and
calculate the estimate of the derivative of ƒ φ (θ)=e −2πi(θ)φ ƒ(θ) with respect to the first control parameter through the summation
f
φ
′
(
θ
0
)
=
∑
s
∈
S
d
s
q
s
F
s
φ
.
13 . The hybrid computing system of claim 12 , wherein the classical computing system is further configured to calculate the estimate of the derivative of the expectation-value function ƒ′(θ 0 ) from ƒ φ ′(θ 0 ).
14 . The hybrid computing system of claim 12 , wherein the classical computing system is further configured to generate the sub-multi-set of shift values and the sub-multi-set of weighting values according to a targeted probability distribution by:
until a target size N of the sub-multi-set of shift values is reached: randomly sampling a first sampled value s from
{
(
n
-
1
2
)
2
K
❘
n
∈
ℤ
}
according to the targeted probability distribution
P
(
s
)
=
1
(
2
π
Ks
)
2
,
and adding the first sampled value s to the sub-multi-set of shift values S; and
setting all weighting values q s to unity.
15 . The hybrid computing system of claim 12 , wherein the classical computing system is further configured to generate the sub-multi-set of shift values and the sub-multi-set of weighting values according to a targeted probability distribution by:
until a target size N of the sub-multi-set of shift values is reached: randomly sampling a first sampled value s from
{
(
n
-
1
2
)
2
K
❘
n
∈
ℤ
,
n
≥
1
}
,
according to the targeted probability distribution
P
(
s
)
=
2
(
2
π
Ks
)
2
,
and adding the first sampled value s and the additive inverse of the sampled value −s to the sub-multi-set of shift values; and
setting all weighting values q s , q −s to unity.
16 . The hybrid computing system of claim 12 , wherein the classical computing system is further configured to generate the sub-multi-set of shift values and the sub-multi-set of weighting values according to a targeted probability distribution by:
selecting a target size N,
constructing a set of values
S
=
{
(
n
-
1
2
)
2
K
❘
n
∈
ℤ
}
for n=−N+1, . . . , N;
and determining weighting values
q
s
=
1
(
2
π
Ks
)
2
.
17 . The hybrid computing system of claim 12 , wherein the classical computing system is further configured to generate the sub-multi-set of shift values and the sub-multi-set of weighting values according to a targeted probability distribution by:
selecting a target size N,
selecting a positive integer T much smaller than half the target size,
constructing a set of values
S
=
{
(
n
-
1
2
)
2
K
❘
n
∈
ℤ
,
n
≥
1
}
⋂
[
-
(
T
-
1
/
2
)
/
2
K
,
+
(
T
-
1
/
2
)
/
2
K
]
by taking s N s times in such a way as to minimise the variance of the random variable
∑
s
∈
{
(
n
-
1
2
)
/
2
K
❘
n
∈
ℤ
,
n
≥
1
}
⋂
[
-
(
T
-
1
/
2
)
/
2
K
,
+
(
T
-
1
/
2
)
/
2
K
]
(
-
1
)
2
Ks
+
1
2
π
(
2
Ks
)
2
Avg
N
s
(
F
s
)
.
18 . The hybrid computing system of claim 12 , wherein the classical computing system is further configured to pass each trial control parameter value through a parameter-folding function, to modify the value of the trial control parameter value such that no trial control parameter value differs from the first control parameter value by more than a parameter shift limit value.
19 . The hybrid computing system of claim 18 , wherein the parameter folding function has the property τ(x)−x∈p ={pk|k∈ }, where
p
:=
k
γ
,
where p is a real number, k is a positive integer and γ is a is a common divisor of the eigenvalues of the first parameterised quantum evolution.
20 . The hybrid computing system of claim 19 , wherein the parameter-folding function is
τ
(
x
)
=
{
-
c
-
(
(
-
x
)
%
p
)
,
if
x
≤
-
c
-
p
;
x
,
if
x
∈
]
-
c
-
p
,
+
c
+
p
[
;
+
c
+
(
x
%
p
)
,
if
x
≥
+
c
+
p
.
where c is a real number, c=m·p where m is a positive integer, and
where % p stands for the remainder in [0, p[ when dividing by p.