Optimal fermion-to-qubit mapping via ternary trees with applications to reduced quantum states learning

Zhang, Jiang; Kalev, Amir; Mruczkiewicz, Wojciech; Neven, Hartmut

doi:10.22331/q-2020-06-04-276

Cited by 53 publications

(62 citation statements)

References 39 publications

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“…This is a crucial subroutine in many quantum computing applications; see, e.g., Refs. [20,[27][28][29][30][31][32][33]. We present a quantum ML model that uses N Q ¼ OðnÞ copies of ρ to predict expectation values of all n-qubit Pauli observables.…”

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confidence: 99%

“…We highlight this potential by means of an illustrative and practically relevant example: predicting expectation values of Pauli operators in an unknown n-qubit quantum state ρ. This is a central task for many quantum computing applications [20,[27][28][29][30][31][32][33]. To formulate this problem in our framework, suppose the 2n-bit input x specifies one of the 4 n n-qubit Pauli operators P x ∈ fI; X; Y; Zg ⊗n , and suppose that E ρ ðjxihxjÞ prepares the unknown state ρ and maps P x to the fixed observable O, which is then measured; hence,…”

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confidence: 99%

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Information-Theoretic Bounds on Quantum Advantage in Machine Learning

2021

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We study the performance of classical and quantum machine learning (ML) models in predicting outcomes of physical experiments. The experiments depend on an input parameter x and involve execution of a (possibly unknown) quantum process E. Our figure of merit is the number of runs of E required to achieve a desired prediction performance. We consider classical ML models that perform a measurement and record the classical outcome after each run of E, and quantum ML models that can access E coherently to acquire quantum data; the classical or quantum data are then used to predict the outcomes of future experiments. We prove that for any input distribution DðxÞ, a classical ML model can provide accurate predictions on average by accessing E a number of times comparable to the optimal quantum ML model. In contrast, for achieving an accurate prediction on all inputs, we prove that the exponential quantum advantage is possible. For example, to predict the expectations of all Pauli observables in an n-qubit system ρ, classical ML models require 2 ΩðnÞ copies of ρ, but we present a quantum ML model using only OðnÞ copies. Our results clarify where the quantum advantage is possible and highlight the potential for classical ML models to address challenging quantum problems in physics and chemistry.

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Information-Theoretic Bounds on Quantum Advantage in Machine Learning

2021

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“…It is however important to note that the larger k is, the more data is required. More precisely, the number of copies of the state required to infer all the kqubit density operators in an N -qubit system with a given statistical confidence scales exponentially in k [22].…”

Section: Methodsmentioning

confidence: 99%

Experimentally accessible non-separability criteria for multipartite entanglement structure detection

García-Pérez¹,

Kerppo²,

Rossi³

et al. 2021

Preprint

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The description of the complex separability structure of quantum states in terms of partially ordered sets has been recently put forward. In this work, we address the question of how to efficiently determine these structures for unknown states. We propose an experimentally accessible and scalable iterative methodology that identifies, on solid statistical grounds, sufficient conditions for nonseparability with respect to certain partitions. In addition, we propose an algorithm to determine the minimal partitions (those that do not admit further splitting) consistent with the experimental observations. We test our methodology experimentally on a 20-qubit IBM quantum computer by inferring the structure of the 4-qubit Smolin and an 8-qubit W states. In the first case, our results reveal that, while the fidelity of the state is low, it nevertheless exhibits the partitioning structure expected from the theory. In the case of the W state, we obtain very disparate results in different runs on the device, which range from non-separable states to very fragmented minimal partitions with little entanglement in the system. Furthermore, our work demonstrates the applicability of informationally complete POVM measurements for practical purposes on current NISQ devices.

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“…Since its discovery, the Jordan-Wigner transformation [1] and subsequent generalizations [10][11][12][13][14][15][16][17][18][19] have enjoyed great success in probing the fundamental physics of quantum many-body spin models, as well as classical statistical mechanics models through so-called transfer-matrix methods [20][21][22]. An understanding of these mappings has furthermore proven useful for designing fermion-to-qubit mappings with desired properties for simulating fermionic systems on a quantum computer [23][24][25][26][27][28][29][30][31][32]. Here, operator locality in the dual spin model is generally enabled through coupling to an auxiliary gauge field [33][34][35], which endows fermionic-pair excitations with the structure of freely deformable strings on the spin lattice [36].…”

Section: Relation To Prior Workmentioning

confidence: 99%

Free Fermions Behind the Disguise

Elman

Chapman

Flammia

2021

Commun. Math. Phys.

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An invaluable method for probing the physics of a quantum many-body spin system is a mapping to noninteracting effective fermions. We find such mappings using only the frustration graph G of a Hamiltonian H, i.e., the network of anticommutation relations between the Pauli terms in H in a given basis. Specifically, when G is (even-hole, claw)-free, we construct an explicit free-fermion solution for H using only this structure of G, even when no Jordan-Wigner transformation exists. The solution method is generic in that it applies for any values of the couplings. This mapping generalizes both the classic Lieb-Schultz-Mattis solution of the XY model and an exact solution of a spin chain recently given by Fendley, dubbed "free fermions in disguise." Like Fendley's original example, the free-fermion operators that solve the model are generally highly nonlinear and nonlocal, but can nonetheless be found explicitly using a transfer operator defined in terms of the independent sets of G. The associated single-particle energies are calculated using the roots of the independence polynomial of G, which are guaranteed to be real by a result of Chudnovsky and Seymour. Furthermore, recognizing (even-hole, claw)-free graphs can be done in polynomial time, so recognizing when a spin model is solvable in this way is efficient. In a crucial step to proving our result, we additionally prove that there exists a hierarchy of commuting conserved charges for models whose frustration graphs are claw-free only, and hence these models are integrable. Finally, we give several example families of solvable and integrable models for which no Jordan-Wigner solution exists, and we give a detailed analysis of such a spin chain having 4-body couplings using this method.

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Optimal fermion-to-qubit mapping via ternary trees with applications to reduced quantum states learning

Cited by 53 publications

References 39 publications

Information-Theoretic Bounds on Quantum Advantage in Machine Learning

Information-Theoretic Bounds on Quantum Advantage in Machine Learning

Experimentally accessible non-separability criteria for multipartite entanglement structure detection

Free Fermions Behind the Disguise

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