# US 12,443,871 B2 · Scalable tensor-network-based noise mitigation for near-term quantum computing > Agents: this data is available as JSON from the patentgrep API and MCP server — see https://patentgrep.com/llms.txt - Number: US 12,443,871 B2 - Document ID: US12443871B2 - Status: Granted - Kind: B2 - Application number: 18/734,466 - Filed: 2024-06-05 - Published: 2024-12-12 - Granted: 2025-10-14 - Earliest claimed: 2023-06-06 - Assignees: Algorithmiq Oy - Applicants: Algorithmiq Oy - Inventors: Guillermo García Pérez; Sergei Filippov - Web page: https://patentgrep.com/patent/US12443871B2 ## CPC - G06N 10/40: Physical realisations or architectures of quantum processors or components for manipulating qubits, e.g. qubit coupling or qubit control - G06N 10/70: Quantum error correction, detection or prevention, e.g. surface codes or magic state distillation - G06N 3/042: Knowledge-based neural networks; Logical representations of neural networks - G06N: COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS ## Abstract In one aspect, there is provided a noise mitigation method for an execution of a quantum circuit by a quantum processor. In another aspect, there is provided a computing system comprising a quantum processor and a classical computer, the computing system being configured to carry out the method. In another aspect, there is a computer program product including instructions which, when the program is carried out by a computer system comprising a classical computer and a quantum processor, cause the computer system to carry out the method. In another aspect, there is provided a computer program product including instructions which, when the program is carried out by a classical computer, cause the classical computer to carry out the tensor network contractions according to the method. ## Claims ### Claim 1 A noise mitigation method for a noisy physical execution of a quantum circuit by a quantum processor, the method comprising: establishing a model of a noisy physical execution of the quantum circuit by the quantum processor, the model being representative of a noisy physical execution of a unitary operation by the quantum processor which results in a noisy final physical state of N qudits of the quantum processor, the model being represented by a quantum channel, wherein a density operator is at least an approximation of the noisy final physical state of the N qudits of the quantum processor; executing the noisy physical execution of the quantum circuit one or more times by the quantum processor by applying a sequence of G physical operations to the N qudits of the quantum processor to thereby obtain a set of measurement outcomes for each execution of the quantum circuit, wherein each measurement outcome comprises a physical quantum measurement applied to the noisy final physical state of the N qudits after each execution of the quantum circuit by the sequence of G physical operations; deriving a quasi-state tensor network representation of a quasi-state comprising an approximation of the density operator of the noisy final state based on the set of measurement outcomes of the execution of the quantum circuit and a set of dual effects corresponding to the set of measurement outcomes; deriving an observable tensor network representation of a Hermitian operator associated with an observable for a system of N qudits; deriving a noise mitigation tensor network representation of a noise mitigation map, wherein the noise mitigation map is a concatenation of an inverse of the quantum channel followed by a unitary operation, wherein each of the tensor network representations comprises respectively a plurality of quasi-state tensors, observable tensors and noise mitigation tensors; providing the quasi-state tensor network representation, the observable tensor network representation and the noise mitigation tensor network representation to a classical computer; and executing, by the classical computer, a tensor network contraction algorithm to thereby calculate a value of the observable associated with the Hermitian operator for a final quantum state of the N qudits of the quantum circuit, the tensor network contraction algorithm including instructions for contracting a tensor network representation in terms of the quasi-state tensors, the observable tensors and the noise mitigation tensors according to a predetermined contracting rule of physical and virtual indices of the tensors to thereby obtain a noise mitigated value of an expectation value of the observable associated with the Hermitian operator for the final quantum state of the quantum circuit. ### Claim 2 (depends on 1) The method of claim 1, wherein the noisy physical execution of the unitary operation is by an application of a sequence of one or more quantum gates, each quantum gate being described by a gate unitary operation, and wherein establishing the model comprises partitioning the sequence of quantum gates in one or more subsequences of quantum gates, each of the subsequences comprising one or more quantum gates which are executed subsequently and/or simultaneously and being described by a subsequence unitary operation, wherein each subsequence of quantum gates comprises a single quantum gate or a single layer of quantum gates, and wherein establishing the model comprises establishing for each subsequence of quantum gates a subsequence model representative of the noisy physical execution of the subsequence unitary operation by the quantum processor and being described by a subsequence channel. ### Claim 3 (depends on 2) The method of claim 2, wherein establishing a respective subsequence model comprises applying quantum process tomography to the noisy implementation of the respective subsequence of quantum gates described by the subsequence unitary operation by the quantum processor to thereby determined the respective subsequence channel. ### Claim 4 (depends on 2) The method of claim 2, wherein the respective subsequence channel is modelled as the concatenation of the respective subsequence unitary operation and a noise channel representative of the noise in the execution of the respective subsequence of quantum gates by the quantum processor. ### Claim 5 (depends on 2) The method of claim 2, wherein deriving the noise mitigation tensor network representation comprises deriving a first tensor network representation of a first map comprising a plurality of first tensors, deriving for an inverse of each subsequence channel, an inverse channel tensor network representation comprising a plurality of inverse channel tensors, and deriving for each subsequence unitary operation, a subsequence tensor network representation comprising a plurality of subsequence tensors, and wherein deriving the noise mitigation tensor network representation of the noise mitigation map is by an execution of a first tensor network contraction algorithm by the classical computer, the first tensor network contraction algorithm including instructions for contracting the first tensors, inverse channel tensors and subsequence tensors according to a predetermined first contraction rule to thereby obtain the noise mitigation tensors. ### Claim 6 (depends on 5) The method of claim 5, wherein the first tensor network contraction algorithm further comprising a compression of the tensors resulting from the contraction of at least some of the first tensors, the inverse channel tensors and the subsequence tensors to thereby obtain noise mitigation tensors with a bond dimension below a predetermined threshold value. ### Claim 7 (depends on 5) The method of claim 5, wherein the first tensor network contraction algorithm comprises an iterative contraction procedure following a rule for iteratively composing the noise mitigation map from 2K maps in T steps, the rule comprising specifying an initial map M1 comprising one of the 2K maps or a composition of a plurality of the 2K maps, wherein a t-th map Mt, t≥2, is defined according to Mt=Lt∘Mt−1 or Mt=Mt−1∘Rt or Mt=Lt·Mt−1∘Rt, wherein Lt and Rt are one of the 2K maps or a composition of a plurality of the 2K maps such that MT is equal to the noise mitigation map, and wherein the iterative contraction procedure comprises a step of deriving a tensor network representation of M1 comprising the subsequence and/or inverse channel tensors of the 2K maps whose composition results in M1, and T−1 iterative steps, wherein in the t-th iterative step, t=1, . . . , T−1, a (t+1)-th tensor network representation of Mt+1 is derived by contracting the tensor network representations of M, and the tensor network representation of respective ones of Rt+1 and Lt+1 according to a (t+1)-th contraction rule to thereby obtain a plurality of (t+1)-th tensors of the (t+1)-th tensor network representation. ### Claim 8 (depends on 7) The method of claim 7, wherein for at least one iterative step, the first tensor network contraction algorithm comprises instruction for compressing the (t+1)-th tensors such that their bond dimension is below a predetermined threshold value. ### Claim 9 (depends on 8) The method of claim 8, wherein compressing the tensors comprises an execution of a truncation algorithms or an application of a variational algorithm. ### Claim 10 (depends on 7) The method of claim 7, wherein the tensor network representation of Mt+1 is derived by contracting the tensor network representation of Mt according to the (t+1)-th contraction rule. ### Claim 11 (depends on 1) The method of claim 1, wherein at least one tensor network representation comprises at least one tensor associated with each qudit, and a single tensor associated with each qudit, wherein the tensor B[k] associated with the k-th qudit has at least one physical index and at least one virtual index. ### Claim 12 (depends on 1) The method of claim 1, wherein for at least one operator or map Q, the respective tensor network representation is a matrix product operator representation, Q=Σa1, . . . ,aN=0X−1Aa0[0]⊗Aa0a1[1]⊗ . . . ⊗AaN−3aN−2[N−2]⊗AaN−2[N−1], wherein for any values of the virtual indicies ak−1, ak the map Aak−1ak[k] is acting on the k-th qudit, the map Aa0[0] is acting on the 0-th qubit and the map AaN−2[N−1] is acting on the (N−1)-th qubit, and wherein χ is the bond dimension. ### Claim 13 (depends on 1) The method of claim 1, wherein for at least one operator or map, the operator or map of the respective tensor network representation is a projected entangled pair operator representation or a tree tensor network representation. ### Claim 14 (depends on 1) The method of claim 1, wherein the effects are L-producible, L≥1, i.e., each effect is a tensor product of operators each of which is acting on at most L qudits, i.e., Πm(k0, . . . , kN−1)=Πk0(0)⊗Πk1(1)⊗ . . . ⊗Πkr−1(r−1), wherein Πkn(n) acts on at most L qudits. ### Claim 15 (depends on 1) The method of claim 1, wherein deriving at least one of the tensor network representations is at least partially carried out by an algorithm executed by the classical computer. ### Claim 16 A computing system comprising a quantum processor and a classical computer, the computing system being configured to establish a model of a noisy physical execution of a quantum circuit by the quantum processor, the model being representative of a noisy physical execution of a unitary operation U by the quantum processor which results in a noisy final physical state of N qudits of the quantum processor, the model being represented by a quantum channel, wherein a density operator is at least an approximation of the noisy final physical state of the N qudits of the quantum processor; execute the noisy physical execution of the quantum circuit one or more times by the quantum processor by applying a sequence of G physical operations to the N qudits of the quantum processor to thereby obtain a set of measurement outcomes for each execution of the quantum circuit, wherein each measurement outcome comprises a physical quantum measurement applied to the noisy final physical state of the N qudits after each execution of the quantum circuit by the sequence of G physical operations; derive a quasi-state tensor network representation of a quasi-state comprising an approximation of the density operator of the noisy final state based on the set of measurement outcomes of the execution of the quantum circuit and a set of dual effects corresponding to the set of measurement outcomes; derive an observable tensor network representation of a Hermitian operator which is associated with an observable for a system of N qudits; derive a noise mitigation tensor network representation of a noise mitigation map, wherein the noise mitigation map is a concatenation of an inverse of the quantum channel followed by a unitary operation, wherein each of the tensor network representations comprises respectively a plurality of quasi-state tensors, observable tensors and noise mitigation tensors; provide the quasi-state tensor network representation, the observable tensor network representation and the noise mitigation tensor network representation to a classical computer; and execute, by the classical computer, a tensor network contraction algorithm to thereby calculate a value of the observable associated with the Hermitian operator for a final quantum state of the N qudits of the quantum circuit, the tensor network contraction algorithm including instructions for contracting a tensor network representation of in terms of the quasi-state tensors, the observable tensors and the noise mitigation tensors according to a predetermined contracting rule of physical and virtual indices of the tensors to thereby obtain a noise mitigated value of an expectation value of the observable associated with the Hermitian operator for the final quantum state of the quantum circuit. ### Claim 17 A non-transitory computer program product including instructions which, when the program is carried out by a computer system comprising a classical computer and a quantum processor, cause the computer system to: establish a model of a noisy physical execution of a quantum circuit by the quantum processor, the model being representative of a noisy physical execution of a unitary operation by the quantum processor which results in a noisy physical final state of N qudits of the quantum processor, the model being represented by a quantum channel, wherein a density operator is at least an approximation of the noisy final physical state of the N qudits of the quantum processor; execute the noisy physical execution of the quantum circuit one or more times by the quantum processor by applying a sequence of G physical operations to the N qudits of the quantum processor to thereby obtain a set of measurement outcomes for each execution of the quantum circuit, wherein each measurement outcome comprises a physical quantum measurement applied to the noisy final physical state of the N qudits after each execution of the quantum circuit by the sequence of G physical operations; derive a quasi-state tensor network representation of a quasi-state comprising an approximation of the density operator of the noisy final state based on the set of measurement outcomes of the execution of the quantum circuit and a set of dual effects corresponding to the set of measurement outcomes; derive an observable tensor network representation of a Hermitian operator which is associated with an observable for a system of N qudits; derive a noise mitigation tensor network representation of a noise mitigation map, wherein the noise mitigation map is a concatenation of an inverse of the quantum channel ε followed by a unitary operation, wherein each of the tensor network representations comprises respectively a plurality of quasi-state tensors, observable tensors and noise mitigation tensors; provide the quasi-state tensor network representation, the observable tensor network representation and the noise mitigation tensor network representation to a classical computer; and execute, by the classical computer, a tensor network contraction algorithm to thereby calculate a value of the observable associated with the Hermitian operator for a final quantum state of the N qudits of the quantum circuit, the tensor network contraction algorithm including instructions for contracting a tensor network representation in terms of the quasi-state tensors, the observable tensors and the noise mitigation tensors according to a predetermined contracting rule of physical and virtual indices of the tensors to thereby obtain a noise mitigated value of an expectation value of the observable associated with the Hermitian operator for the final quantum state of the quantum circuit. ### Claim 18 (depends on 16) The computing system of claim 16, wherein: the noisy physical execution of the unitary operation is by an application of a sequence of one or more quantum gates, each quantum gate being described by a gate unitary operation, and wherein establishing the model comprises partitioning the sequence of quantum gates in one or more subsequences of quantum gates, each of the subsequences comprising one or more quantum gates which are executed subsequently and/or simultaneously and being described by a subsequence unitary operation, wherein each subsequence of quantum gates comprises a single quantum gate or a single layer of quantum gates, and wherein establishing the model comprises establishing for each subsequence of quantum gates a subsequence model representative of the noisy physical execution of the subsequence unitary operation by the quantum processor and being described by a subsequence channel; and deriving the noise mitigation tensor network representation comprises deriving a first tensor network representation of a first map comprising a plurality of first tensors, deriving for the inverse of each subsequence channel, an inverse channel tensor network representation comprising a plurality of inverse channel tensors, and deriving for each subsequence unitary operation, a subsequence tensor network representation comprising a plurality of subsequence tensors, and wherein deriving the noise mitigation tensor network representation of the noise mitigation map is by an execution of a first tensor network contraction algorithm by the classical computer, the first tensor network contraction algorithm including instructions for contracting the first tensors, inverse channel tensors and subsequence tensors according to a predetermined first contraction rule to thereby obtain the noise mitigation tensors. ### Claim 19 (depends on 18) The computing system of claim 18, wherein the first tensor network contraction algorithm comprises an iterative contraction procedure following a rule for iteratively composing the noise mitigation map from 2K maps in T steps, the rule comprising specifying an initial map M1 comprising one of the 2K maps or a composition of a plurality of the 2K maps, wherein a t-th map Mt, t≥2, is defined according to Mt=Lt∘Mt−1 or Mt=Mt−1∘R, or Mt=Lt∘Mt−1∘Rt, wherein Lt and Rt are one of the 2K maps or a composition of a plurality of the 2K maps such that MT is equal to the noise mitigation map, and wherein the iterative contraction procedure comprises a step of deriving a tensor network representation of M1 comprising the subsequence and/or inverse channel tensors of the 2K maps whose composition results in M1, and T−1 iterative steps, wherein in the t-th iterative step, t=1, . . . , T−1, a (t+1)-th tensor network representation of Mt+1 is derived by contracting the tensor network representations of M, and the tensor network representation of respective ones of Rt+1 and Lt+1 according to a (t+1)-th contraction rule to thereby obtain a plurality of (t+1)-th tensors of the (t+1)-th tensor network representation. ### Claim 20 (depends on 19) The computing system of claim 19, wherein the tensor network representation of Mt+1 is derived by contracting the tensor network representation of Mt according to the (t+1)-th contraction rule. ### Claim 21 (depends on 17) The computer program product of claim 17, wherein: the noisy physical execution of the unitary operation is by an application of a sequence of one or more quantum gates, each quantum gate being described by a gate unitary operation, and wherein establishing the model comprises partitioning the sequence of quantum gates in one or more subsequences of quantum gates, each of the subsequences comprising one or more quantum gates which are executed subsequently and/or simultaneously and being described by a subsequence unitary operation, wherein each subsequence of quantum gates comprises a single quantum gate or a single layer of quantum gates, and wherein establishing the model comprises establishing for each subsequence of quantum gates a subsequence model representative of the noisy physical execution of the subsequence unitary operation by the quantum processor and being described by a subsequence channel; and deriving the noise mitigation tensor network representation comprises deriving a first tensor network representation of a first map comprising a plurality of first tensors, deriving for the inverse of each subsequence channel, an inverse channel tensor network representation comprising a plurality of inverse channel tensors, and deriving for each subsequence unitary operation, a subsequence tensor network representation comprising a plurality of subsequence tensors, and wherein deriving the noise mitigation tensor network representation of the noise mitigation map is by an execution of a first tensor network contraction algorithm by the classical computer, the first tensor network contraction algorithm including instructions for contracting the first tensors, inverse channel tensors and subsequence tensors according to a predetermined first contraction rule to thereby obtain the noise mitigation tensors. ### Claim 22 (depends on 21) The computing program product of claim 21, wherein the first tensor network contraction algorithm comprises an iterative contraction procedure following a rule for iteratively composing the noise mitigation map from 2K maps in T steps, the rule comprising specifying an initial map M1 comprising one of the 2K maps or a composition of a plurality of the 2K maps, wherein a t-th map Mt, t≥2, is defined according to Mt=Lt∘Mt−1 or Mt=Mt−1∘Rt or Mt=Lt∘Mt−1∘Rt, wherein Lt and Rt are one of the 2K maps or a composition of a plurality of the 2K maps such that M is equal to the noise mitigation map, and wherein the iterative contraction procedure comprises a step of deriving a tensor network representation of M1 comprising the subsequence and/or inverse channel tensors of the maps whose composition results in M1, and T−1 iterative steps, wherein in the t-th iterative step, t=1, . . . , T−1, a (t+1)-th tensor network representation of Mt+1 is derived by contracting the tensor network representations of Mt and the tensor network representation of respective ones of Rt+1 and Lt+1 according to a (t+1)-th contraction rule to thereby obtain a plurality of (t+1)-th tensors of the (t+1)-th tensor network representation. ### Claim 23 (depends on 22) The computing program product of claim 22, wherein the tensor network representation of Mt+1 is derived by contracting the tensor network representation of Mt according to the (t+1)-th contraction rule.