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arXiv:1707.06739 · Physical Review A

Experimental demonstration of selective quantum process tomography on an NMR quantum information processor

Akshay Vijay Gaikwad, Diksha Rehal, Amandeep Singh, Arvind, Kavita Dorai

Published Feb 7, 2018• 36 citations
arXivHTMLPDF

Abstract

We present the NMR implementation of a scheme for selective and efficient quantum process tomography without ancilla. We generalize this scheme such that it can be implemented efficiently using only a set of measurements involving product operators. The method allows us to estimate any element of the quantum process matrix to a desired precision, provided a set of quantum states can be prepared efficiently. Our modified technique requires fewer experimental resources as compared to the standard implementation of selective and efficient quantum process tomography, as it exploits the special nature of NMR measurements to allow us to compute specific elements of the process matrix by a restrictive set of subsystem measurements. To demonstrate the efficacy of our scheme, we experimentally tomograph the processes corresponding to ``no operation,'' a controlled-NOT (CNOT), and a controlled-Hadamard gate on a two-qubit NMR quantum information processor, with high fidelities.

DOI
10.1103/physreva.97.022311

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We present the NMR implementation of a scheme for selective and efficient quantum process tomography without ancilla. We generalize this scheme such that it can be implemented efficiently using only a set of measurements involving product operators. The method allows us to estimate any element of the quantum process matrix to a desired precision, provided a set of quantum states can be prepared efficiently. Our modified technique requires fewer experimental resources as compared to the standard implementation of selective and efficient quantum process tomography, as it exploits the special nature of NMR measurements to allow us to compute specific elements of the process matrix by a restrictive set of subsystem measurements. To demonstrate the efficacy of our scheme, we experimentally tomograph the processes corresponding to ``no operation,'' a controlled-NOT (CNOT), and a controlled-Hadamard gate on a two-qubit NMR quantum information processor, with high fidelities.
Alternatively to the full reconstruction of an unknown quantum process, the so-called selective and efficient quantum process tomography (SEQPT) allows estimating, individually and up to the required accuracy, a given element of the matrix that describes such an operation with a polynomial amount of resources. The implementation of this protocol has been carried out with success to characterize the evolution of a quantum system that is well described by a trace-preserving quantum map. Here, we deal with a more general type of quantum process that does not preserve the trace of the input quantum state, which naturally arises in the presence of imperfect devices and system-environment interactions, in the context of quantum information science or quantum dynamics control. In that case, we show with the aid of a priori information on the losses structure of the quantum channel that the SEQPT reconstruction can be adapted to reconstruct the non-trace-preserving map. We explicitly describe how to implement the reconstruction in an arbitrary Hilbert space of finite dimension ddd. The method is experimentally verified on a superconducting quantum processor provided by IBM Quantum services, by estimating several non-trace-preserving quantum processes in dimensions up to d=6d=6d=6. Our results show that it is possible to efficiently reconstruct non-trace-preserving processes, with high precision, and with significantly higher fidelity than when the process is assumed to be trace preserving.
The experimental implementation of selective quantum process tomography (SQPT) involves computing individual elements of the process matrix with the help of a special set of states called quantum 2-design states. However, the number of experimental settings required to prepare input states from quantum 2-design states to selectively and precisely compute a desired element of the process matrix is still high, and hence constructing the corresponding unitary operations in the lab is a daunting task. In order to reduce the experimental complexity, we mathematically reformulated the standard SQPT problem, which we term the modified SQPT (MSQPT) method. We designed the generalized quantum circuit to prepare the required set of input states and formulated an efficient measurement strategy aimed at minimizing the experimental cost of SQPT. We experimentally demonstrated the MSQPT protocol on the IBM QX2 cloud quantum processor and selectively characterized various two- and three-qubit quantum gates.
Alternatively to the full reconstruction of an unknown quantum process, the so-called selective and efficient quantum process tomography (SEQPT) allows estimating, individually and up to the required accuracy, a given element of the matrix that describes such an operation with a polynomial amount of resources. The implementation of this protocol has been carried out with success to characterize the evolution of a quantum system that is well described by a trace preserving quantum map. Here, we deal with a more general type of quantum process that does not preserve the trace of the input quantum state, which naturally arises in the presence of imperfect devices and system-environment interactions, in the context of quantum information science or quantum dynamics control. In that case, we show that with the aid of {\it a priori} information on the losses structure of the quantum channel, the SEQPT reconstruction can be adapted to reconstruct the non-trace-preserving map. We explicitly describe how to implement the reconstruction in an arbitrary Hilbert space of finite dimension ddd. The method is experimentally verified on a superconducting quantum processor of the IBM Quantum services, by estimating several non trace-preserving quantum processes in dimensions up to d=6d=6d=6. Our results show that it is possible to efficiently reconstruct non trace-preserving processes, with high precision, and with significantly higher fidelity than when the process is assumed to be trace-preserving.
Abstract Quantum process tomography is an effective method for measuring the quantum channel parameters in the field of quantum information. However, standard quantum process tomography (SQPT) cannot obtain arbitrarily specific process–matrix elements because it requires a global reconstruction algorithm. Direct quantum tomography can be used to obtain specific matrix elements of interest without the need for a global reconstruction algorithm. Therefore, direct process quantum tomography (DQPT) has attracted widespread attention. DQPT based on weak measurements has been recently proposed. However, additional auxiliary states and post‐selection processes are necessary in these schemes, which increases the complexity of practical experiments and reduces the utilization of measurement resources. In this study, a DQPT protocol without the assistance of pointer states is proposed, which reduces the complexity of practical experiments. Subsequently, the scheme is generalized to arbitrary qubit systems and experimentally perform the protocol on a nuclear magnetic resonance (NMR) system provided by a quantum cloud platform. Finally, considering that when the qubit resources are insufficient, a superimposed coherent state is used as the pointer state to complete the direct characterization of a quantum process. The measurement protocol is easily scalable and integrated, laying the foundation for direct tomography on a chip.
The temporal evolution of a quantum system can be characterized by quantum process tomography, a complex task that consumes a number of physical resources scaling exponentially with the number of subsystems. An alternative approach to the full reconstruction of a quantum channel is the selective and efficient quantum process tomography, a method that allows estimating, individually and up to the required accuracy, each element of the matrix that describes the process, using only a polynomial amount of resources. The implementation of this protocol is closely related to the possibility of building a complete set of mutually unbiased bases (MUBs), whose existence is known only when the dimension of the Hilbert space is the power of a prime number. However, an extension of the method that uses tensor products of maximal sets of MUBs has been recently introduced. Here we explicitly describe how to implement the algorithm for a selective and efficient estimation of a quantum process in a nonprime power dimension and conduct, an experimental verification of the method in a Hilbert space of dimension d=6d=6d=6. This is the smallest space for which a complete set of MUBs is not known to exist, but it can be decomposed as a tensor product of two Hilbert spaces of dimensions D1=2{D}_{1}=2D1​=2 and D2=3{D}_{2}=3D2​=3, in which a complete set of MUBs is already known. The six-dimensional states were codified in the discretized transverse linear momentum of photons. The state preparation and detection stages are dynamically programed with the use of only-phase spatial light modulators, in a versatile experimental setup that allows one to implement the algorithm in any finite dimension.
Quantum shadow tomography based on the classical shadow representation provides an efficient way to estimate properties of an unknown quantum state without performing a full quantum state tomography. In scenarios where estimating the expectation values for only certain classes of observables is required, obtaining information about the entire density matrix is unnecessary. We propose a partial quantum shadow tomography protocol that estimates a subset of density matrix elements relevant to the expectation values of structured observables. Specifically, we identify specific subsets of the tomographically complete set Cl(2)⊗n{\mathrm{Cl}}(2)^{\otimes n}Cl(2)⊗n and a simple pseudo-inverse of the associated channel, which can be used to estimate all elements of the density matrix with the same active order. By restricting the protocol to smaller subsets of single-qubit Pauli measurements, it becomes experimentally more efficient. We demonstrate the advantage over unitary designs, such as the Clifford, full Pauli basis, and methods utilizing mutually unbiased bases, by analytically deriving error bounds and numerically evaluating the protocol on structured operators. We experimentally demonstrate the partial shadow estimation scheme for a wide class of two-qubit states (pure, entangled, and mixed) in the nuclear magnetic resonance (NMR) platform. The full density matrix, reconstructed experimentally by combining different partial estimators, achieves fidelities around 99%.