DETAILED ACTION
Notice of Pre-AIA or AIA Status
The present application, filed on or after March 16, 2013, is being examined under the first inventor to file provisions of the AIA .
Claim Rejections - 35 USC § 103
The following is a quotation of 35 U.S.C. 103 which forms the basis for all obviousness rejections set forth in this Office action:
A patent for a claimed invention may not be obtained, notwithstanding that the claimed invention is not identically disclosed as set forth in section 102, if the differences between the claimed invention and the prior art are such that the claimed invention as a whole would have been obvious before the effective filing date of the claimed invention to a person having ordinary skill in the art to which the claimed invention pertains. Patentability shall not be negated by the manner in which the invention was made.
Claims 1-6 and 9-12 are rejected under 35 U.S.C. 103 as being unpatentable over
Pierre ROUCHON et al. (hereinafter ROUCHON) US 2024/0303519 A1 [Foreign Priority: EP 23305313.1 Filed 2023-03-08],
in view of M. Cerezo et.al. (hereinafter Cerezo) Variational Quantum Fidelity Estimation, arXiv:1906.09253v2 [quant-ph] 3 Mar 2020,
Regarding claim 1:
ROUCHON discloses:
A method for determining an approximated final quantum state in such a way that a fidelity between the approximated final quantum state and an exact final quantum state is greater than or equal to a lower bound, the method comprising:
[0066]:
The method of physically performing an operation on an open quantum system comprises providing a desired final state of a density operator ρ of the open quantum system after an operation to be performed on a known initial state of the open quantum system. The method further comprises: (i) using the method or device of the second aspect to provide an approximated final state of the density operator, wherein said initial state of said density operator ρ models the known initial state of the open quantum system and wherein said Lindblad master equation defining the dynamics models the evolution of the system under the operation; (ii) calculating a fidelity between the approximated final state and desired final state; (iii) comparing the calculated fidelity to a threshold value; (iv) performing: (a) if the fidelity is above the threshold value, extraction of the parameters of said Lindblad master equation corresponding to physically controllable parameters of the operation; or (b) if the fidelity is below the threshold value, repetition of steps (i)-(iii) with a modified Lindblad master equation with modified parameters corresponding to physically controllable parameters of the operation until the calculated fidelity is above the threshold value, and extracting the modified parameters of the modified Lindblad master equation resulting in the calculated fidelity being above the threshold value, and extracting the parameters corresponding to physically controllable parameters of the operation of the modified Lindblad master equation corresponding to the calculated fidelity being above the threshold value; and (v) physically performing the operation on the open quantum system using the extracted parameters
[BRI: the threshold is the “lower bound”]
receiving an initial quantum state of a plurality of qubits, the initial quantum state being in a matrix product representation comprising a product of tensors;
[0199]:
For an open quantum system including one or more entities, it is thus possible to determine a propagator P as a function of the matrix F, a second-order approximation of which can be computed with limited computational resources. This computation takes advantage of both the local character of the nominal dynamics and the fact that the perturbation is decomposable into a finite sum of tensor products of local operators.
[0200]:
The simulation of an open quantum system including a plurality of quantum entities is particularly appropriate to simulate a quantum logic gate involving two or more qubits
receiving a quantum circuit comprising quantum gates to be applied successively to the initial quantum state, wherein each quantum gate among the quantum gates is a single-qubit quantum gate or a two-qubit quantum gate;
[0281]:
Just as it is possible, as previously discussed, to simulate a quantum logic gate for two or more qubits, it is also possible to simulate a quantum logic gate for a single qubit, for instance a Z-gate
[0336]:
The CNOT gate allows to entangle two qubits called “control” and “target” and is used in particular in the quantum circuit of the repetition code in order to extract the parity of two adjacent data qubits thanks to an auxiliary qubit called “ancilla” by carrying out two successive CNOTs. The CNOT gate can be written as:
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50
380
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wherein
I
a
,
b
and
Z
a
,
b
are the Pauli operators acting on the cat-qubit code space respectively for the first “control” and second “target” cat-qubits and
X
b
is the Pauli operator acting on the cat-qubit code space of the target cat-qubit.
a Z-gate.
defining a lower bound of a fidelity between the approximated final quantum state and the exact final quantum state;
[0020]:
The processor is further arranged to determine a propagator P as a time product of the matrices F(k) for approximating a final state ρ.sub.f of the density operator ρ based on an initial state ρ.sub.0 of said density operator ρ.
[0066]:
initial state of said density operator ρ models the known initial state of the open quantum system and wherein said Lindblad master equation defining the dynamics models the evolution of the system under the operation; (ii) calculating a fidelity between the approximated final state and desired final state; (iii) comparing the calculated fidelity to a threshold value; (iv) performing: (a) if the fidelity is above the threshold value, extraction of the parameters of said Lindblad master equation corresponding to physically controllable parameters of the operation
[BRI: the threshold defines the lower bound]
iterating over the quantum gates of the quantum circuit in an intended order of application
[0359]:
the methods or devices of either the first or second aspects may be used to guide the choice of experimental parameters to be used in an operation on an open quantum system formed from quantum states (e.g. comprising only stabilized quantum entities, or both stabilized and non-stabilized quantum entities). For instance, a desired operation may be to start from a known initial state ρ(0) of the open quantum system and subsequently evolve the system under a physically (e.g. experimentally) controlled interaction to a desired (and thus also known) final state of the density operator ρ.
[0359]:
a first choice of the Lindblad master equation may be input to derive an approximated final state of the density operator. Said approximated (i.e. simulated) final state may be compared with the desired final state. If the fidelity between the approximated final state and the desired final state is below a certain threshold (i.e. indicating that the approximated final state does not suitably match up with the desired final state), then a second choice of the Lindblad master equation may be input to derive a second approximated final state of the density operator, which is subsequently compared with the desired final state. This method may be iterated until the fidelity between the approximated final state and the desired final state is above a certain threshold. The choice of the Lindblad master equation which results in the approximated final state yielding the acceptable fidelity (i.e. above the certain threshold) dictates the choice of experimental parameters to be used. That is, the parameters of the Lindblad master equation yielding the acceptable fidelity are extracted, and an operation on an open quantum system is physically performed using the extracted parameters.
[0107]:
Lindblad master equation can first be seen as a sum of a Hamiltonian part, which models the unitary part of the dynamics, corresponding to a closed evolution, and a dissipative part, which corresponds to the finite sum expressed with the Lindblad operators and models the interaction of the quantum system with its environment.
[BRI: in the idealized case of a closed quantum system, the unitary part of the dynamics is mathematically equivalent to applying a sequence of quantum gates in the intended order to the initial state, with each gate corresponding to a unitary operator in the product
U
f
i
n
a
l
=
(
U
n
, …
U
2
,
U
1
)
target truncation fidelities defined for truncation of diagonal matrices in relation to two-qubit quantum gates among the quantum gates of the quantum circuit not yet applied to the initial quantum state, is greater than or equal to said lower bound;
[0045]:
the sum over index j is truncated at a threshold such that the difference between the penultimate and final summed terms is smaller than 10.sup.−8, and where Tr( ) denotes the trace;
[0044]:
determining a propagator P by numerically integrating
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over the time period T;
[0045]:
deriving an approximated final state of the density operator ρ by applying said propagator P to the initial state ρ (0) of said density operator ρ of the open quantum system.
[0062]:
The index j may be truncated at a threshold such that the difference between the between the penultimate and final summed terms is at machine precision of the conventional computer.
[0303]:
The sum over index j is in practice truncated at a finite number such that the difference between two successive iterations (i.e. difference between the penultimate and final summed terms) is smaller than a chosen threshold. For instance, the threshold may be on the order of machine precision of the conventional computer performing the simulation, e.g., such as between
10
-
8
and
10
-
16
. For instance, the threshold may be
10
-
12
.
[BRI: A target truncation fidelity is the fidelity between the exact state (before truncation) and the truncated state (after truncation) for a given gate application. The goal is to ensure that the truncation does not degrade the state’s fidelity below a specified threshold. The not yet applied” refers to the quantum gates in the circuit that have not yet been executed on the current (intermediate) quantum state during the simulation or approximation process. Truncated at a threshold such that the difference between the penultimate (second-to-last) and final summed terms is less than
10
-
8
, essentially applying a numerical truncation criterion to ensure the omitted terms contribute negligibly to the total. Tr( · ) denotes the trace of a matrix (or operator), that supports comparing the truncated fidelity (e.g., a fidelity matrix after truncation) to a lower bound, the condition can be written as: Tr(
ρ
t
r
u
n
c
) ≥ lower bound (threshold) where
ρ
t
r
u
n
c
is the density matrix (or operator) after truncation. The trace here is the sum of the eigenvalues (or diagonal elements) of the truncated object]
in a next iteration, using the updated quantum state as the initial quantum state
[0359]:
Lindblad master equation may be input to derive a second approximated final state of the density operator, which is subsequently compared with the desired final state. This method may be iterated until the fidelity between the approximated final state and the desired final state is above a certain threshold.
[BRI: in an iterative Lindblad master equation approach, the updated quantum state can be used as the initial state for the next iteration, provided the iteration is designed to refine the approximation toward the desired final state. In fact, the iteration is tracking for the desired state that is above the threshold]
defining an approximated final quantum state being equal to the updated quantum state of a last iteration
[0359]:
Lindblad master equation may be input to derive a second approximated final state of the density operator, which is subsequently compared with the desired final state. This method may be iterated until the fidelity between the approximated final state and the desired final state is above a certain threshold.
[BRI: in this iterative Lindblad master equation approach, the approximated final quantum state is defined as the updated quantum state at the iteration where the fidelity with the desired state exceeds the threshold]
ROUCHON does not explicitly disclose:
applying a current quantum gate of the quantum gates of the quantum circuit to the initial quantum state to obtain an updated quantum state.
if the current quantum gate is a two-qubit quantum gate, factorizing a portion of the updated quantum state resulting from the application of the current quantum gate to the initial quantum state by singular value decomposition into a product of a complex unitary matrix, a diagonal matrix having a diagonal of non-negative real numbers, and an adjoint complex unitary matrix;
truncating a bond dimension of the diagonal matrix to a target bond dimension being determined in such a way that a product of:
a truncation fidelity of the diagonal matrix after truncation, truncation fidelities of truncated diagonal matrices of previous iterations, wherein each of said truncated diagonal matrices of previous iterations is obtained by truncating a diagonal matrix of a previous iteration obtained by singular value decomposition of a portion of an updated quantum state of a previous iteration resulting from the application of a two-qubit quantum gate of a previous iteration to an initial quantum state of a previous iteration, and
However, Cerezo discloses:
applying a current quantum gate of the quantum gates of the quantum circuit to the initial quantum state to obtain an updated quantum state.
[Proposition 6, Page 15]:
we are interested in bounding since F (
ρ
,
σ
) is the desired fidelity while
F
^
(
ρ
m
'
,
σ
) is the quantity that VQFE actually outputs,
[Proposition 6, Page 15]:
we will need to re-formulate this as a probabilistic bound since B3 depends on the estimate
T
^
of a random variable. To do this re-formulation, let us consider the sources of statistical noise on
T
^
. In the first step of VQFE, we call the VQSD subroutine to approximately diagonalize
ρ
_ up to some error bounded by C =
D
H
S
(
ρ
,
ρ
'
).
Here,
PNG
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21
167
media_image3.png
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‘
where {
r
i
'
} and {|
r
i
'
} are approximations of the eigenvalues and eigenvectors of
ρ
respectively. While
ρ
'
can only be estimated up to some finite sampling precision, such error can be tuned appropriately.
[BRI: In VQE-style algorithms, the approximation is produced by a parametrized quantum circuit whose gates are applied sequentially to an initial quantum state, updating the state at each step and this matches the description of applying a current quantum gate to obtain an updated quantum state. Each quantum gate in the circuit transforms the quantum state, and the classical optimizer tunes parameters to minimize error relative to the target observable].
if the current quantum gate is a two-qubit quantum gate, factorizing a portion of the updated quantum state resulting from the application of the current quantum gate to the initial quantum state by singular value decomposition into a product of a complex unitary matrix, a diagonal matrix having a diagonal of non-negative real numbers, and an adjoint complex unitary matrix;
[7, Page 5]:
Problem 1. (Low-rank Fidelity Estimation): Input: Two poly(n)-sized quantum circuit descriptions U and V that prepare n-qubit states ρ and σ on a subset of qubits all initially in the |0⟩ state,
[3, Page 3]:
A non-zero cost C was obtained by applying a random unitary close to the identity to the diagonal form of ρ.
[BRI: In linear algebra, an invertible complex square matrix U is unitary if its matrix inverse U equals its conjugate transpose U which is the identity matrix].
[Abstract, Page 1]:
To compute our bounds, we introduce a hybrid quantum-classical algorithm, called
Variational Quantum Fidelity Estimation, that in volves three steps: (1) variationally diagonalize ρ, (2) compute matrix elements of σ in the eigenbasis of ρ, and (3) combine these matrix elements to compute our bounds.
[Appendix, Page 10]:
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993
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160
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[1, Page 2]:
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Figure 1: The VQFE algorithm. (1) First, ρ is diagonalized with a hybrid quantum-classical optimization loop, out putting the largest eigenvalues {ri} of ρ and a gate sequence that prepares the associated eigenvectors. (2) Second, a hybrid quantum-classical computation gives the matrix elements of σ in the eigenbasis of ρ. (3) Finally, classical processing gives upper and lower bounds on F( ρ, σ)
[3, Page 2]:
Figure 1 shows the overall structure of the VQFE algorithm. VQFE involves three steps: (1) ap proximately diagonalize ρ with a variational hybrid quantum-classical algorithm, (2) compute matrix elements of σ in the eigenbasis of ρ, (3) classically process these matrix elements to produce certified bounds on F(ρ ,σ). The first subroutine employs Variational Quantum State Diagonalization (VQSD) [18], a variational hybrid algorithm that takes in two copies of ρ and
[3, Page 3]:
outputs approximations of the m-largest eigenvalues {
r
i
} and a gate sequence U that prepares the associated eigenvectors {|
r
i
⟩}. This subroutine involves a quantum-classical optimization loop that minimizes a cost function C that quantifies how far
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24
110
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from being diagonal.
[BRI: a subroutine optimization loop is an iteration, because it is a control structure that repeatedly executes a block of code (in this case, the subroutine call or related operations) under certain conditions until a stopping criterion is met]
truncating a bond dimension of the diagonal matrix to a target bond dimension being determined in such a way that a product of:
[1, Page 1]:
Our bounds are based on the truncated fidelity, which involves evaluating (1) for σ and
ρ
m
, a truncated version of ρ obtained by projecting ρ onto the subspace associated with its m-largest eigenvalues.
[2, Page 2]:
Proposition 2. The truncated fidelity F(
ρ
m
,
σ
m
ρ
) is monotonically increasing in m, and the truncated generalized fidelity
F
*
(
ρ
m
,
σ
m
ρ
) is monotonically decreasing in m. Ultimately, we will consider the case when ρ is either low rank or ϵ-low-rank. Here we define the ϵ-rank as a generalization of the rank to within some ϵ error:
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507
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where d is the Hilbert space dimension.
[BRI: within the “truncated fidelity” framework, projecting a density matrix onto its m-largest eigenvalues is mathematically equivalent to truncating its bond dimension to a target value, and this projection defines a subspace in which the fidelity bounds are evaluated. The projection onto the m-largest eigenvectors defines a subspace of dimension m in Hilbert space]
a truncation fidelity of the diagonal matrix after truncation, truncation fidelities of truncated diagonal matrices of previous iterations, wherein each of said truncated diagonal matrices of previous iterations is obtained by truncating a diagonal matrix of a previous iteration obtained by singular value decomposition of a portion of an updated quantum state of a previous iteration resulting from the application of a two-qubit quantum gate of a previous iteration to an initial quantum state of a previous iteration, and
[1, Page 2]:
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132
429
media_image6.png
Greyscale
Figure 1: The VQFE algorithm. (1) First, ρ is diagonalized with a hybrid quantum-classical optimization loop, out putting the largest eigenvalues {ri} of ρ and a gate sequence that prepares the associated eigenvectors. (2) Second, a hybrid quantum-classical computation gives the matrix elements of σ in the eigenbasis of ρ. (3) Finally, classical processing gives upper and lower bounds on F( ρ, σ)
[3, Page 2]:
Figure 1 shows the overall structure of the VQFE algorithm. VQFE involves three steps: (1) ap proximately diagonalize ρ with a variational hybrid quantum-classical algorithm, (2) compute matrix elements of σ in the eigenbasis of ρ, (3) classically process these matrix elements to produce certified bounds on F(ρ ,σ). The first subroutine employs Variational Quantum State Diagonalization (VQSD) [18], a variational hybrid algorithm that takes in two copies of ρ and
[3, Page 3]:
outputs approximations of the m-largest eigenvalues {
r
i
} and a gate sequence U that prepares the associated eigenvectors {|
r
i
⟩}. This subroutine involves a quantum-classical optimization loop that minimizes a cost function C that quantifies how far
PNG
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24
110
media_image7.png
Greyscale
from being diagonal.
[BRI: a subroutine optimization loop is an iteration, because it is a control structure that repeatedly executes a block of code (in this case, the subroutine call or related operations) under certain conditions until a stopping criterion is met]
[7, Page 5]:
Problem 1. (Low-rank Fidelity Estimation): Input: Two poly(n)-sized quantum circuit descriptions U and V that prepare n-qubit states ρ and σ on a subset of qubits all initially in the |0⟩ state,
[3, Page 3]:
A non-zero cost C was obtained by applying a random unitary close to the identity to the diagonal form of ρ.
[BRI: In linear algebra, an invertible complex square matrix U is unitary if its matrix inverse U equals its conjugate transpose U which is the identity matrix].
[Abstract, Page 1]:
To compute our bounds, we introduce a hybrid quantum-classical algorithm, called
Variational Quantum Fidelity Estimation, that in volves three steps: (1) variationally diagonalize ρ, (2) compute matrix elements of σ in the eigenbasis of ρ, and (3) combine these matrix elements to compute our bounds.
[Appendix, Page 10]:
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73
993
media_image4.png
Greyscale
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30
160
media_image5.png
Greyscale
[1, Page 2]:
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media_image6.png
132
429
media_image6.png
Greyscale
Figure 1: The VQFE algorithm. (1) First, ρ is diagonalized with a hybrid quantum-classical optimization loop, out putting the largest eigenvalues {ri} of ρ and a gate sequence that prepares the associated eigenvectors. (2) Second, a hybrid quantum-classical computation gives the matrix elements of σ in the eigenbasis of ρ. (3) Finally, classical processing gives upper and lower bounds on F( ρ, σ)
[3, Page 2]:
Figure 1 shows the overall structure of the VQFE algorithm. VQFE involves three steps: (1) ap proximately diagonalize ρ with a variational hybrid quantum-classical algorithm, (2) compute matrix elements of σ in the eigenbasis of ρ, (3) classically process these matrix elements to produce certified bounds on F(ρ ,σ). The first subroutine employs Variational Quantum State Diagonalization (VQSD) [18], a variational hybrid algorithm that takes in two copies of ρ and
[3, Page 3]:
outputs approximations of the m-largest eigenvalues {
r
i
} and a gate sequence U that prepares the associated eigenvectors {|
r
i
⟩}. This subroutine involves a quantum-classical optimization loop that minimizes a cost function C that quantifies how far
PNG
media_image7.png
24
110
media_image7.png
Greyscale
from being diagonal.
[BRI: a subroutine optimization loop is an iteration, because it is a control structure that repeatedly executes a block of code (in this case, the subroutine call or related operations) under certain conditions until a stopping criterion is met]
in a next iteration, using the updated quantum state as the initial quantum state.
[Proposition 6, Page 15]:
we are interested in bounding since F(
ρ
,
σ
) is the desired fidelity while
F
^
(
ρ
m
'
,
σ
) is the quantity that VQFE actually outputs,
[Proposition 6, Page 15]:
we will need to re-formulate this as a probabilistic bound since B3 depends on the estimate
T
^
of a random variable. To do this re-formulation, let us consider the sources of statistical noise on
T
^
. In the first step of VQFE, we call the VQSD subroutine to approximately diagonalize
ρ
up to some error bounded by C =
D
H
S
(
ρ
,
ρ
'
).
Here,
PNG
media_image3.png
21
167
media_image3.png
Greyscale
‘
where {
r
i
'
} and {|
r
i
'
} are approximations of the eigenvalues and eigenvectors of
ρ
respectively. While
ρ
'
can only be estimated up to some finite sampling precision, such error can be tuned appropriately.
[BRI: Perhaps known to a POSITA that a VQSD subroutine represents “itration”. In the VQSD-based diagonalization framework, the updated quantum state from one iteration is typically used as the initial state for the next iteration, enabling faster convergence and more accurate diagonalization]
It would be obvious to one of ordinary skill in the art before the effective filing date of the present application to combine ROUCHON, and Cerezo.
ROUCHON teaches fidelity between quantum states with ensuring that the fidelity is greater than lower bound.
Cerezo teaches truncated fidelity for quantum states and diagonal matrix computations.
One of ordinary skills would be to combine ROUCHON, and Cerezo that can bound that increase monotonically with eigenvalues (Cerezo 1, Page 1]).
Regarding claim 2:
ROCHON does not disclose explicitly:
wherein truncating the diagonal matrix comprises canceling one or more diagonal elements of the diagonal matrix
However, Cerezo discloses:
wherein truncating the diagonal matrix comprises canceling one or more diagonal elements of the diagonal matrix.
[Abstract, 1]
Computing quantum state fidelity will be important to verify and characterize states pre pared on a quantum computer. In this work, we propose novel lower and upper bounds for the fidelity F(ρ, σ) based on the” truncacated fidelity” F(
ρ
m
, σ), which is evaluated for a state
ρ
m
obtained by projecting ρ onto its m-largest eigenvalues.
To compute our bounds, we introduce a hybrid quantum-classical algorithm, called Variational Quantum Fidelity Estimation, that in volves three steps: (1) variationally diagonalize ρ, (2) compute matrix elements of σ in the eigenbasis of ρ, and (3) combine these matrix elements to compute our bounds
[Proposition 6, Page 16]
Let ρ and σ be quantum states, and suppose we have access to a subroutine that diagonalizes ρ up to error bounded by
C = DHS(ρ, ρ′).
Let m and ζ be parameters. Then, VQFE runs in time O(m6/ζ2) and outputs an additive ±γ-approximation of F(ρ, σ), for
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42
757
media_image9.png
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where ϵ is determined by m =
r
a
n
k
e
(ρ).
This proposition is very similar to Lemma 6 except that γ′ is replaced by γ
[BRI: within the variational quantum fidelity estimation framework , the lower and upper bounds are computed from a truncated fidelity F (
ρ
m
,
σ), where
ρ
m
is obtained by projecting the original state ρ onto its m largest eigenvalues. Mathematically, if ρ is diagonal in its eigenbasis, the truncated state computed with all eigenvalues (diagonal entries) beyond the m-th largest are set to zero. This is exactly the operation of “removing” or “zeroing” the smaller diagonal elements of the diagonal matrix representing ρ in its eigenvalues providing “cancel” those elements
It would be obvious to one of ordinary skill in the art before the effective filing date of the present application to combine ROUCHON, and Cerezo.
ROUCHON teaches fidelity between quantum states with ensuring that the fidelity is greater than lower bound.
Cerezo teaches truncated fidelity for quantum states and diagonal matrix computations.
One of ordinary skills would be to combine ROUCHON, and Cerezo that can bound that increase monotonically with eigenvalues (Cerezo 1, Page 1]).
Regarding claim 3:
ROUCHON does not explicitly disclose:
wherein said one or more diagonal elements are canceled in increasing order, starting from an element with a smallest value.
However, Cerezo discloses:
[Abstract, 1]
Computing quantum state fidelity will be important to verify and characterize states prepared on a quantum computer. In this work, we propose novel lower and upper bounds for the fidelity F(ρ, σ) based on the” truncated fidelity” F(
ρ
m
,σ), which is evaluated for a state
ρ
m
obtained by projecting ρ onto its m-largest eigenvalues. Our bounds can be refined, i.e., they tighten monotonically with m. To compute our bounds, we introduce a hybrid quantum-classical algorithm, called Variational Quantum Fidelity Estimation, that in volves three steps: (1) variationally diagonalize ρ, (2) compute matrix elements of σ in the eigenbasis of ρ, and (3) combine these matrix elements to compute our bounds,
[Proposition 6, Page 16]
Let ρ and σ be quantum states, and suppose we have access to a subroutine that diagonalizes ρ up to error bounded by
C = DHS(ρ, ρ′).
Let m and ζ be parameters. Then, VQFE runs in time O(m6/ζ2) and outputs an additive ±γ-approximation of F(ρ, σ), for
PNG
media_image9.png
42
757
media_image9.png
Greyscale
where ϵ is determined by m =
r
a
n
k
e
(ρ).
This proposition is very similar to Lemma 6 except that γ′ is replaced by γ
[BRI: Mathematically, if ρ is diagonal in its eigenbasis, the truncated state computed with all eigenvalues (diagonal entries) beyond the m-th largest are set to zero. This is exactly the operation of “removing” or “zeroing” the smaller diagonal elements of the diagonal matrix representing ρ in its eigenvalues providing “cancel” those elements. Under certain conditions (e.g., small off-diagonal entries, symmetric structure), cancelling diagonal elements in increasing order can lead to monotonically tighter bounds on the largest eigenvalue. This is a common trick in numerical linear algebra for deflating or for stabilization]
Regarding claim 4:
ROCHOUN and Castrillo do not explicitly disclose:
wherein the target bond dimension is determined in such a way that a truncation fidelity of the diagonal matrix truncated to said target bond dimension is equal to or greater than a target truncation fidelity defined for said diagonal matrix, and a number of diagonal elements canceled from the diagonal matrix when truncating the bond dimension of the diagonal matrix to the target bond dimension is maximized.
However, Cerezo discloses:
wherein the target bond dimension is determined in such a way that a truncation fidelity of the diagonal matrix truncated to said target bond dimension is equal to or greater than a target truncation fidelity defined for said diagonal matrix, and a number of diagonal elements canceled from the diagonal matrix when truncating the bond dimension of the diagonal matrix to the target bond dimension is maximized.
[Abstract, 1]
Computing quantum state fidelity will be important to verify and characterize states prepared on a quantum computer. In this work, we propose novel lower and upper bounds for the fidelity F( ρ, σ) based on the” truncated fidelity” F(
ρ
m
,
σ), which is evaluated for a state
ρ
m
obtained by projecting ρ onto its m-largest eigenvalues. Our bounds can be refined, i.e., they tighten monotonically with m. To compute our bounds, we introduce a hybrid quantum-classical algorithm, called Variational Quantum Fidelity Estimation, that in volves three steps: (1) variationally diagonalize ρ, (2) compute matrix elements of σ in the eigenbasis of ρ, and (3) combine these matrix elements to compute our bounds.
[Proposition 6, Page 16]:
Let ρ and σ be quantum states, and suppose we have access to a subroutine that diagonalizes ρ up to error bounded by
C = DHS(ρ, ρ′).
Let m and ζ be parameters. Then, VQFE runs in time O(m6/ζ2) and outputs an additive ±γ-approximation of F(ρ,σ), for
PNG
media_image9.png
42
757
media_image9.png
Greyscale
where ϵ is determined by m =
r
a
n
k
e
(ρ).
This proposition is very similar to Lemma 6 except that γ′ is replaced by γ
[1, Page 1]:
Verification and characterization of these mixed states will be important, and hence efficient algorithms will be needed for this purpose. A widely used measure for verification and characterization is the fidelity
[1, Page 1]:
For example, one may be interested in the fidelity with a fixed target state (i.e., for
verification) or the fidelity between subsystems of many body states to study behavior near a phase transition (i.e., for characterization).
[4, Page 4]:
Figure 2 shows our VQFE implementations on IBM’s quantum computer simulator. The left and right panels show representative results for n = 3 and n =6 qubits, respectively,
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392
507
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[4, Page 4]:
We chose σ as a random state and
⊗
j
=
1
n
ρ
j
as a tensor product state, where the latter can be diagonalized with a depth-one quantum circuit ansatz. As one can see, as m increases the TFB rapidly converge to F( ρ, σ), and since C is small (∼ 10−6), the TFB can be viewed as bounds on F(ρ, σ).
[Proposition 5, Page 13]:
(Low-Rank Fidelity Estimation) via the Choi-Jamiołkowski isomorphism over the unitary channels,
PNG
media_image11.png
58
806
media_image11.png
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where D(
H
d
) is the space of d × d dimensional hermitian matrices. Consider now the 2n-qubit maximally entangled state,
PNG
media_image12.png
73
734
media_image12.png
Greyscale
where j = (
j
1
,
j
2
, …
j
n
) is a binary vector taking values
j
k
in {0,1}, and where E is an efficient unitary entangling gate (e.g., a depth-two circuit composed of Hadamard and CNOT gates), where |0⟩ = |0⟩⊗2n. A special case of Low-Rank Fidelity Estimation is when ρ and σ correspond to the Choi states of ˜U and ˜V. In this case, as the input to Low-Rank Fidelity Estimation, we would be given the gate sequences U=(
U
~
⊗
1
) E and V=(
V
~
⊗
1
) E
which respectively prepare the pure states ρ and σ as
PNG
media_image13.png
82
741
media_image13.png
Greyscale
Then, the fidelity between ρ and σ is given by
PNG
media_image14.png
49
756
media_image14.png
Greyscale
[BRI: In this construction, the bond dimension of the diagonal matrix in the MPS/PEPS representation is directly tied to the bond dimension of the maximally entangled resource state.
Regarding claim 5:
ROUCHON does not explicitly disclose:
wherein the target truncation fidelity is defined as:
PNG
media_image15.png
37
139
media_image15.png
Greyscale
, where
F
r
a
n
d
o
m
the lower bound and where
N
g
is a number of two-qubit quantum gates of the quantum circuit.
However, Cerezo discloses:
wherein the target truncation fidelity is defined as:
PNG
media_image15.png
37
139
media_image15.png
Greyscale
, where
F
r
a
n
d
o
m
the lower bound and where
N
g
is a number of two-qubit quantum gates of the quantum circuit.
[5, Page 4]:
Let m∗ denote the minimum value of m needed for our bounds to become tighter than the SSFB. Figure 3 plots m∗ for the TFB, CCFB, and CTIB for systems with n = 2, ...,7 qubits. The results were obtained by averaging m∗ over 2000 random states ρ and σ. We considered two cases of interest: when ρ is a low rank state, and when it has an exponentially decaying spectrum leading to full rank but high purity.
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418
510
media_image16.png
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[5, Page 4]:
The TFB also provide information regarding the closeness of eigenvectors (e.g., upper-TFB which are ≈ 1 for m = 1,2), and can detect level crossings where the structure of the subspace spanned by {|
r
i
⟩} drastically changes. For instance, near h = 1 the m = 3, 4 TFB present a discontinuity from the crossing between a uniform eigenstate and a pair of exactly degenerate non-uniform symmetry-breaking states. Hence, the fidelity spectrum provides information about the structure of the states beyond the scope of the SSFB or even F(ρ, σ).
[BRI: TFB is a truncated fidelity bound. When a quantum state’s density matrix ρ has a low rank (few non-zero eigenvalues) and its spectrum decays exponentially, the state is highly pure in the sense that most of its weight is concentrated on a small number of basis states. With Low rank + exponential decay leads to high purity in support but small overlap with arbitrary target states. The fidelity decay is exponential in the number of qubits if the decay rate is per-qubit and decay rate α depends on the physical mechanism causing the exponential suppression (e.g., decoherence, noise, or structure in the state). In summary, for a low-rank state with exponentially decaying spectrum, the fidelity to a generic target state can decay exponentially with the number of qubits, even though the state is full rank]
Regarding claim 6:
ROUCHON does not explicitly disclose:
wherein in a first iteration, the target truncation fidelity is defined as:
PNG
media_image15.png
37
139
media_image15.png
Greyscale
where
F
r
a
n
d
o
m
the lower bound and where
N
g
is a number of two-qubit quantum gates of the quantum circuit, in subsequent iterations, the target truncation fidelity is updated depending on truncation fidelities of diagonal matrices of previous iterations or kept constant.
However, Cerezo discloses:
wherein in a first iteration, the target truncation fidelity is defined as:
PNG
media_image15.png
37
139
media_image15.png
Greyscale
where
F
r
a
n
d
o
m
the lower bound and where
N
g
is a number of two-qubit quantum gates of the quantum circuit, in subsequent iterations, the target truncation fidelity is updated depending on truncation fidelities of diagonal matrices of previous iterations or kept constant.
[1, Page 2]:
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media_image6.png
132
429
media_image6.png
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Figure 1: The VQFE algorithm. (1) First, ρ is diagonalized with a hybrid quantum-classical optimization loop, out putting the largest eigenvalues {ri} of ρ and a gate sequence that prepares the associated eigenvectors. (2) Second, a hybrid quantum-classical computation gives the matrix elements of σ in the eigenbasis of ρ. (3) Finally, classical processing gives upper and lower bounds on F( ρ, σ)
[3, Page 2]:
Figure 1 shows the overall structure of the VQFE algorithm. VQFE involves three steps: (1) ap proximately diagonalize ρ with a variational hybrid quantum-classical algorithm, (2) compute matrix elements of σ in the eigenbasis of ρ, (3) classically process these matrix elements to produce certified bounds on F(ρ ,σ). The first subroutine employs Variational Quantum State Diagonalization (VQSD) [18], a variational hybrid algorithm that takes in two copies of ρ and
[3, Page 3]:
outputs approximations of the m-largest eigenvalues {
r
i
} and a gate sequence U that prepares the associated eigenvectors {|
r
i
⟩}. This subroutine involves a quantum-classical optimization loop that minimizes a cost function C that quantifies how far
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media_image7.png
24
110
media_image7.png
Greyscale
from being diagonal.
[BRI: a subroutine optimization loop is an iteration, because it is a control structure that repeatedly executes a block of code (in this case, the subroutine call or related operations) under certain conditions until a stopping criterion is met]
It would be obvious to one of ordinary skill in the art before the effective filing date of the present application to combine ROUCHON, and Cerezo.
ROUCHON teaches fidelity between quantum states with ensuring that the fidelity is greater than lower bound.
Cerezo teaches truncated fidelity for quantum states and diagonal matrix computations.
One of ordinary skills would be to combine ROUCHON, Cerezo and Chen that can bound that increase monotonically with eigenvalues (Cerezo 1, Page 1]).
Regarding claim 9:
ROUCHON discloses:
determining an actual lower bound of the fidelity between the approximated final quantum state and an exact final quantum state defined as a product of the truncation fidelities.
[0066]:
calculating a fidelity between the approximated final state and desired final state; (iii) comparing the calculated fidelity to a threshold value; (iv) performing: (a) if the fidelity is above the threshold value, extraction of the parameters of said Lindblad master equation corresponding to physically controllable parameters of the operation; or (b) if the fidelity is below the threshold value
[0066]:
physically performing the operation on the open quantum system using the extracted parameters.
[0125]
open quantum system has a Hilbert space H. The Hilbert space is well known to the skilled person in the field of quantum mechanics as it makes it possible to represent, with an infinite dimensional space, the set of quantum states of a quantum system. The Hilbert space can therefore be qualified as a state-space. Similarly, each quantum entity of the open quantum system has a respective Hilbert space
H
q
, such that:
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media_image17.png
19
89
media_image17.png
Greyscale
where:
⊗
denotes the tensor product.
[BRI: Hilbert space where each quantum entity has its own Hilbert space and the total space is their tensor product, the fidelity between the approximated final state and the exact final state is determined by (and bounded below by) the product of the truncation fidelities of the individual subsystems, with equality when the state is a product state and truncations are independent]
Regarding claim 10:
ROUCHON discloses:
further comprising, having defined the approximated final quantum state and having determined the lower bound of the fidelity: outputting the approximated final quantum state and the actual lower bound of the fidelity.
[0066]:
calculating a fidelity between the approximated final state and desired final state; (iii) comparing the calculated fidelity to a threshold value; (iv) performing: (a) if the fidelity is above the threshold value, extraction of the parameters of said Lindblad master equation corresponding to physically controllable parameters of the operation; or (b) if the fidelity is below the threshold value,
[0239]:
Referring to FIG. 10, the case in which the open quantum system comprises a single quantum entity is now considered. This case allows both to understand the mechanisms of the previous formulas and to highlight that it is possible to push the approximation of the function F to an arbitrary order to simulate an open quantum system,
[0247]:
The r-order parts
F
r
as well as the Hermitian trace-class operators are determined by a recurrence process on r, recalling that the zero-order part F.sup.0 is already known and corresponds to the identity matrix and that the Hermitian trace-class operators S.sub.d.sup.0 are also known since they form the orthonormal Hermitian basis of the decoherence-free space D.sub.0,
[BRI: Recalling zero-order parts of fidelity is an reconstruction of an approximation (zeroth-order term) to process fidelity]
[0268]:
In an operation 330, The processor 5 determines if the Q order has been reached, i.e. whether the Q-order part of the matrix F has already been determined. If not, the value of r is incremented in an operation 340 and the previous operations 300, 310 and 320 are implemented again for the next order. If the order Q has been reached, the process of FIG. 10 stops and the matrix F is outputted.
[BRI: when the order Q is reached and outputting matrix F means the process matrix has been reconstructed the process matrix up to the chosen truncation, which Encodes the best estimate of the quantum channel (and thus the predicted final state for any input), and provides a fidelity estimate between the ideal and experimental process, which is a lower bound due to statistical limits]
Regarding claim 11:
ROUCHON discloses:
further comprising, having defined the approximated final quantum state and having determined the lower bound of the fidelity: deciding based on the actual lower bound of the fidelity and on a predetermined criterion whether the approximated final quantum state is a
realistic approximation of the exact final quantum state.
[0066]:
calculating a fidelity between the approximated final state and desired final state; (iii) comparing the calculated fidelity to a threshold value; (iv) performing: (a) if the fidelity is above the threshold value, extraction of the parameters of said Lindblad master equation corresponding to physically controllable parameters of the operation; or (b) if the fidelity is below the threshold value
[0066]:
physically performing the operation on the open quantum system using the extracted parameters.
[BRI: Perhaps known to a POSITA that in quantum information practice, comparing calculated fidelity to a predetermined threshold is a standard method for deciding whether an approximated quantum for finding the acceptable approximation of the target state]
Regarding claim 12:
ROUCHON discloses:
A non-transitory computer readable storage medium, having stored thereon a computer program comprising program instructions, the computer program being loadable into a data processing unit and adapted to cause the data-processing unit to carry out a method of claim 1.
[0034]:
the present invention according to the first aspect relates to a computer-readable storage medium comprising the computer program stored thereon,
[0063]:
There is also provided a computer program or computer-readable data carrier comprising instructions which, when executed by a conventional computer, causes the conventional computer to carry of the according to the second aspect.
[0064:
the method or device of the second aspect may be further used to determine: (i) the choice of experimental parameters to be used in an operation on an open quantum system formed from quantum states; and/or (ii) the choice of design parameters for the one or more quantum entities of an open quantum system.
Claims 7-8 are rejected under 35 U.S.C. 103 as being unpatentable over
Pierre ROUCHON et al. (hereinafter ROUCHON) US 2024/0303519 A1 [Foreign Priority: EP 23305313.1 Filed 2023-03-08],
in view of M. Cerezo et.al. (hereinafter Cerezo) Variational Quantum Fidelity Estimation, arXiv:1906.09253v2 [quant-ph] 3 Mar 2020,
further in view of Ranyiliu Chen et.al. (hereinafter Chen) Variational Quantum Algorithms for Trace Distance and Fidelity Estimation, arXiv:2012.05768v3 [quant-ph], 11 Nov 2021.
Regarding claim 7:
ROCHOUN and Cerezo do not explicitly disclose:
wherein, if the initial quantum state and the approximated final quantum state are a pure quantum states, the truncation fidelity is defined as:
PNG
media_image18.png
86
257
media_image18.png
Greyscale
where
A
i
i
are elements of the diagonal matrix before truncation and
A
'
i
i
are elements of the diagonal matrix after truncation.
However, Chen discloses:
wherein, if the initial quantum state and the approximated final quantum state are a pure quantum states, the truncation fidelity is defined as:
PNG
media_image18.png
86
257
media_image18.png
Greyscale
where
A
i
i
are elements of the diagonal matrix before truncation and
A
'
i
i
are elements of the diagonal matrix after truncation.
[I, Page 1]:
When at least one of the states is pure, the task of fidelity estimation reduces to the simple case of calculating the square root of the state overlap F(ρ,σ) = √ Trρσ which can be obtained by the Swap test [32].
[III, Page 5]:
we introduce Variational Fidelity Estimation (VFE) as a hybrid quantum-classical algorithm for estimating fidelity in the most general case where two mixed states are provided
[III B, Page 6]:
The task of quantum state learning is to find the correct unitary operation such that one could prepare any target mixed state ρ from an initialized state (usually |0 0| on each qubit).
[I, 1]:
the authors variationally estimate the truncated fidelity via a hybrid classical-quantum algorithm. The truncated fidelity bounds the exact fidelity and is a good approximation of it when one of the states is known to have a low rank.
[Supplemental Material, Page 11]:
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535
1059
media_image19.png
Greyscale
As the equality of Eq S3 holds when
H
A
B
~
is diagonalized under the computational basis, the proof is complete.
[I, Page 1]:
the fidelity F(ρ, σ) are defined as follows:
PNG
media_image20.png
106
465
media_image20.png
Greyscale
where
.
1
denotes the trace norm. When at least one of the states is pure, the task of fidelity estimation reduces to the simple case of calculating the square root of the state overlap
PNG
media_image21.png
24
161
media_image21.png
Greyscale
which can be obtained by the Swap-test [32]. Thus the trace distance in this case is bounded by 1- F (
ρ
,
σ
)
≤
D (
ρ
,
σ
)
≤
√
1-
F
(
ρ
,
σ
)
2
.
[BRI: Perhaps as known to a POSITA the claim limitation equation, that in linear algebra, it is the squared correlation between the two vectors, often called the overlap or cosine similarity. Perhaps known to a POSITA that it relates to the quantum fidelity as defined for mixed state and that can be reduced to
PNG
media_image22.png
35
149
media_image22.png
Greyscale
using the following:
PNG
media_image23.png
69
332
media_image23.png
Greyscale
which is the squared overlap of the two state vectors. By expanding a mixed state in a basis and represent it as a vector of probabilities (or amplitudes), the fidelity between two such states can be expressed in terms of inner products of these vectors and when the states are diagonal in the same basis one can take the element-wise product of their probability vectors, the fidelity formula reduces to the cosine-squared form above. Perhaps known to a POSITA that spectral decomposition can provide the diagonal elements (eigenvalues) of a matrix both before and after truncation, because truncation is simply a change in the size of the matrix, and the decomposition is computed for the given size.
H
A
B
a
n
d
H
A
B
~
are the matrices before and after truncation includes the corresponding represented diagonal elements of
A
i
i
a
n
d
A
'
i
i
]
It would be obvious to one of ordinary skills in the art before the effective filing date of the present application to combine ROUCHON, Cerezo and Chen.
ROUCHON teaches fidelity between quantum states with ensuring that the fidelity is greater than lower bound.
Cerezo teaches truncated fidelity for quantum states and diagonal matrix computations.
Chen teaches target truncation fidelities defined for truncation of diagonal matrices in relation to two-qubit quantum gates.
One of ordinary skills would be to combine ROUCHON, Cerezo and Chen that estimate the fidelity by optimizing over unitary on an ancillary system (Chen [Abstract, Page 1].
Regarding claim 8:
ROUCHON, and Cerezo do not explicitly disclose:
wherein, if the initial quantum state and the approximated final quantum state are mixed quantum states, the truncation fidelity is defined as:
PNG
media_image18.png
86
257
media_image18.png
Greyscale
where
A
i
i
are elements of the diagonal matrix before truncation and
A
'
i
i
are elements of the diagonal matrix after truncation.
However, Chen discloses:
wherein, if the initial quantum state and the approximated final quantum state are mixed quantum states, the truncation fidelity is defined as:
PNG
media_image18.png
86
257
media_image18.png
Greyscale
where
A
i
i
are elements of the diagonal matrix before truncation and
A
'
i
i
are elements of the diagonal matrix after truncation.
[III, Page 5]:
we introduce Variational Fidelity Estimation (VFE) as a hybrid quantum-classical algorithm for estimating fidelity in the most general case where two mixed states are provided
[III B, Page 6]:
The task of quantum state learning is to find the correct unitary operation such that one could prepare any target mixed state ρ from an initialized state (usually |0 0| on each qubit).
[I, 1]:
In [34], the authors variationally estimate the truncated fidelity via a hybrid classical-quantum algorithm. The truncated fidelity bounds the exact fidelity and is a good approximation of it when one of the states is known to have a low rank.
[Supplemental Material, Page 11]:
PNG
media_image19.png
535
1059
media_image19.png
Greyscale
As the equality of Eq S3 holds when
H
A
B
~
is diagonalized under the computational basis, the proof is complete.
[I, Page 1]:
the fidelity F(ρ, σ) are defined as follows:
PNG
media_image20.png
106
465
media_image20.png
Greyscale
where
.
1
denotes the trace norm. When at least one of the states is pure, the task of fidelity estimation reduces to the simple case of calculating the square root of the state overlap
PNG
media_image21.png
24
161
media_image21.png
Greyscale
which can be obtained by the Swap-test [32]. Thus the trace distance in this case is bounded by 1- F (
ρ
,
σ
)
≤
D (
ρ
,
σ
)
≤
√
1-
F
(
ρ
,
σ
)
2
.
[BRI: Perhaps as known to a POSITA the claim limitation equation, that in linear algebra, it is the squared correlation between the two vectors, often called the overlap or cosine similarity. Perhaps known to a POSITA that it relates to the quantum fidelity as defined for mixed state and that can be reduced to
PNG
media_image22.png
35
149
media_image22.png
Greyscale
using the following:
PNG
media_image23.png
69
332
media_image23.png
Greyscale
which is the squared overlap of the two state vectors. By expanding a mixed state in a basis and represent it as a vector of probabilities (or amplitudes), the fidelity between two such states can be expressed in terms of inner products of these vectors and when the states are diagonal in the same basis one can take the element-wise product of their probability vectors, the fidelity formula reduces to the cosine-squared form above. Perhaps known to a POSITA that spectral decomposition can provide the diagonal elements (eigenvalues) of a matrix both before and after truncation, because truncation is simply a change in the size of the matrix, and the decomposition is computed for the given size.
H
A
B
a
n
d
H
A
B
~
are the matrices before and after truncation includes the corresponding represented diagonal elements of
A
i
i
a
n
d
A
'
i
i
]
It would be obvious to one of ordinary skills in the art before the effective filing date of the present application to combine ROUCHON, Cerezo and Chen.
ROUCHON teaches fidelity between quantum states with ensuring that the fidelity is greater than lower bound.
Cerezo teaches truncated fidelity for quantum states and diagonal matrix computations.
Chen teaches target truncation fidelities defined for truncation of diagonal matrices in relation to two-qubit quantum gates.
One of ordinary skills would be to combine ROUCHON, Cerezo and Chen that estimate the fidelity by optimizing over unitary on an ancillary system (Chen [Abstract, Page 1].
Conclusion
Any inquiry concerning this communication or earlier communications from the
examiner should be directed to TIRUMALE KRISHNASWAMY RAMESH whose telephone number is (571)272-4605. The examiner can normally be reached by phone.
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If attempts to reach the examiner by telephone are unsuccessful, the examiner’s supervisor, Li B Zhen can be reached on phone (571-272-3768). The fax phone number for the organization where this application or proceeding is assigned is 571-273-8300.
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/TIRUMALE K RAMESH/Examiner, Art Unit 2121
/Li B. Zhen/Supervisory Patent Examiner, Art Unit 2121
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