Quantum tomography is an important tool for the characterisation of quantum operations. In this paper, we present a framework of quantum tomography in fermionic systems. Compared with qubit systems, fermions obey the superselection rule, which sets constraints on states, processes and measurements in a fermionic system. As a result, we can only partly reconstruct an operation that acts on a subset of fermion modes, and the full reconstruction always requires at least one ancillary fermion mode in addition to the subset. We also report a protocol for the full reconstruction based on gates in Majorana fermion quantum computer, including a set of circuits for realising the informationally-complete state preparation and measurement.
Quantum tomography is an important tool for the characterisation of quantum operations. In this paper, we present a framework of quantum tomography in fermionic systems. Compared with qubit systems, fermions obey the superselection rule, which sets constraints on states, processes and measurements in a fermionic system. As a result, we can only partly reconstruct an operation that acts on a subset of fermion modes, and the full reconstruction always requires at least one ancillary fermion mode in addition to the subset. We also report a protocol for the full reconstruction based on gates in Majorana fermion quantum computer, including a set of circuits for realising the informationally-complete state preparation and measurement.
Quantum tomography is an important tool for the characterization of quantum operations. In this paper we present a framework of quantum tomography in fermionic systems. Compared with qubit systems, fermions obey the superselection rule, which sets constraints on states, processes, and measurements in a fermionic system. As a result, we can only partly reconstruct an operation that acts on a subset of fermion modes and the full reconstruction always requires at least one ancillary fermion mode in addition to the subset. We also report a protocol for the full reconstruction based on gates in a Majorana-fermion quantum computer, including a set of circuits for realizing the informationally complete state preparation and measurement.
PACS. 03.65.Ta – Foundations of quantum mechanics; measurement theory. PACS. 03.65.Wj – State reconstruction, quantum tomography. Abstract. – We propose a method for the tomographic reconstruction of qubit states for a general class of solid-state systems in which the Hamiltonians are represented by spin operators, e.g., with Heisenberg-, XXZ-, or XY-type exchange interactions. We analyze the implementation of the projective operator measurements, or spin measurements, on qubit states. All the qubit states for the spin Hamiltonians can be reconstructed by using experimental data. Quantum information processing requires the effective measurement of quantum states. However, a single quantum measurement can only obtain partial information of a quantum state. The reconstruction of a quantum state requires measuring a complete set of observables on an ensemble of identically prepared copies of the system. This method called quantum state tomography [1], is very important because any unknown state can be ascertained by tomographic measurements. Moreover, the full description of qubit states can increase the accuracy of quantum operations. Tomographic measurements have been experimentally implemented for, e.g., the nuclear spin state of an NMR system [2], the electromagnetic field and photon
Quantum state tomography is a technique in quantum information science used to reconstruct the density matrix of an unknown quantum state, providing complete information about the quantum state. It is of significant importance in fields such as quantum computation, quantum communication, and quantum simulation. However, as the size of the quantum system increases, the number of measurement settings and sampling requirements for quantum state tomography grow exponentially with the number of qubits. This not only makes experimental design and implementation more complex, but also exacerbates the consumption of experimental resources. These limitations severely hinder the application of state tomography in large-scale quantum systems. To reduce measurement settings and improve sampling efficiency, this study proposes a state tomography method based on auxiliary systems. This method can be implemented through either entanglement between the quantum system to be measured and a quantum auxiliary system or through correlation between the quantum system and a probabilistic classical auxiliary system. Measurements on the entire joint system enable more efficient extraction of information about the quantum state to be measured. This method relies on standard quantum gate operations and requires only two measurement settings, with a total sampling complexity of O(d2), significantly simplifying experimental operations and measurement processes. Additionally, this study provides two schemes for measuring purity based on the proposed circuit, one of which achieves measurement precision at the Heisenberg limit. This study validates the effectiveness of the proposed method through a detailed theoretical analysis, a series of numerical simulations, and experiments.
Characterising multi-time quantum processes is essential for analysing temporally correlated noise and for designing effective control and mitigation strategies. A complete operational description through multi-time process tomography requires an informationally complete set of probes, which necessarily includes non-deterministic intermediate operations. On present-day quantum devices, such operations are commonly implemented using mid-circuit measurements and reset, which are technologically limited and can introduce noise and overhead in terms of ancilla requirement. In this work, we study the minimal ancillary dimension required for complete characterisation of multi-time processes. We show that sequential interactions with a single qubit ancilla can generate an informationally complete family of correlated probes for processes of arbitrary length, without requiring mid-circuit measurements or reset. Our result provides a resource-efficient route for complete multi-time process tomography and establishes that one qubit of coherent ancillary memory suffices for full reconstruction of arbitrary multi-time dynamics.
Reconstructing the full quantum state of a many-body system requires the estimation of a number of parameters that grows exponentially with system size. Nevertheless, there are situations in which one is only interested in a subset of these parameters, and a full reconstruction is not needed. A paradigmatic example is a scenario in which one aims at determining all the reduced states only up to a given size. Overlapping tomography provides constructions to address this problem with a number of product measurements much smaller than what is obtained when performing independent tomography of each reduced state. There are, however, many relevant physical systems with a natural notion of locality where one is mostly interested in the reduced states of neighboring particles. In this work, we study this form of local overlapping tomography. First of all, we show that, contrary to its full version, the number of product-measurement settings needed for local overlapping tomography does not grow with system size. Then, we present strategies for qubit and fermionic systems in selected lattice geometries. The developed methods find a natural application in the estimation of many-body systems prepared in current quantum simulators or quantum computing devices, where interactions are often local.
The anticommutation of Fermionic fields raises the problem of simulating the evolution of Fermionic systems by means of commuting quantum systems, say qubits. We tackle the issue considering local Fermionic modes as the elementary systems of an operational probabilistic theory. We show that the locality of Fermionic operations, namely operations on systems that are not causally connected must commute, implies the parity superselection rule, which inhibits the superposition of states with an even and an odd number of excitations. Accordingly we derive the largest probabilistic theory compatible with the parity superselection constraint and show that it lacks two distinctive traits of quantum theory, the local tomography and the monogamy of the entanglement. We generalize the notion of superselection rule to general probabilistic theories as sets of linear constraints on the convex set of states and prove a trade-off between the cardinality of the superselection rule and the degree of holism of the resulting theory. Within this scenario the Fermionic quantum theory and the real quantum theory can be regarded as superselected versions of the usual quantum theory. These results are published in Ref. [7] http://iopscience.iop.org/0295-5075/107/2/20009 and Ref. [8] http://www.worldscientific.com/doi/abs/10.1142/S0217751X14300257. In the last three decades the relation between Fermionic systems and other quantum systems has been throughly investigated from both the computational and the physical point of view. In particular the puzzling anti-commuting nature of the Fermionic systems casts a shadow on the possibility of simulating the physical evolution of a bunch of Fermionic systems by means of commuting quantum systems—say qubits. This issue was raised by R. P. Feynman in 1982 [9], when in his seminal work on physical computation he wondered about the possibility of simulating Fermions by local quantum systems in interaction—what we would call nowadays a quantum computer. The problem is that of encoding the evolution of Fermionic fields onto localized quantum systems. A well-known encoding of N Fermionic systems into N qubits is given by the Jordan-Wigner transform (JWT) [14]. Such an encoding, based on the identification between the Fock space of N Fermions and the Hilbert space of N qubits, provides a -algebra isomorphism between the Fermionic anticommuting algebra and the commuting algebra of qubits. Such a correspondence has been a valuable instrument in modern solid state physics for solving the one dimensional XY spin-chains [18, 15] and then for the understanding of superconductivity and the quantum Hall effect. Moreover, a time-adaptive JWT has been introduced in Ref. [19], which allows to contract Fermionic unitary circuits with the same complexity as for the corresponding spin model. In quantum information science the JWT has been used to extend to the Fermionic case notions as entanglement [1], the entropic area law [24], and universal computation [5]. More recently the JWT, which originally regards one dimensional chains of spin-1=2 systems, has been generalized to any spin [3] and lattice [13] dimension. Despite its computational power, the JWT fails to solve completely the issue established by Feynman: physically local Fermionic operations are mapped into nonlocal quantum ones and viceversa. As noticed by many authors this can lead to ambiguities in defining the partial trace [16, 4, 17, 10], and in assessing