Precision tomography of a three-qubit donor quantum processor in silicon
Mateusz T. Mądzik, Serwan Asaad, Akram Youssry, Benjamin Joecker, Kenneth M. Rudinger, Erik Nielsen, Kevin C. Young, Timothy J. Proctor, Andrew D. Baczewski, Ar… 展开作者
Mateusz T. Mądzik, Serwan Asaad, Akram Youssry, Benjamin Joecker, Kenneth M. Rudinger, Erik Nielsen, Kevin C. Young, Timothy J. Proctor, Andrew D. Baczewski, Arne Laucht, Vivien Schmitt, Fay E. Hudson, Kohei M. Itoh, Alexander M. Jakob, Brett C. Johnson, David N. Jamieson, Andrew S. Dzurak, Christopher Ferrie, Robin Blume-Kohout, Andrea Morello
备注：
51 pages, including supplementary information. v3 reflects the final published version
Nuclear spins were among the first physical platforms to be considered for quantum information processing, because of their exceptional quantum coherence and atomic-scale footprint. However, their full potential for quantum computing has not yet been realized, due to the lack of methods to link nuclear qubits within a scalable device combined with multi-qubit operations with sufficient fidelity to sustain fault-tolerant quantum computation. Here we demonstrate universal quantum logic operations using a pair of ion-implanted 31P donor nuclei in a silicon nanoelectronic device. A nuclear two-qubit controlled-Z gate is obtained by imparting a geometric phase to a shared electron spin, and used to prepare entangled Bell states with fidelities up to 94.2(2.7)%. The quantum operations are precisely characterised using gate set tomography (GST), yielding one-qubit average gate fidelities up to 99.95(2)%, two-qubit average gate fidelity of 99.37(11)% and two-qubit preparation/measurement fidelities of 98.95(4)%. These three metrics indicate that nuclear spins in silicon are approaching the performance demanded in fault-tolerant quantum processors. We then demonstrate entanglement between the two nuclei and the shared electron by producing a Greenberger-Horne-Zeilinger three-qubit state with 92.5(1.0)% fidelity. Since electron spin qubits in semiconductors can be further coupled to other electrons or physically shuttled across different locations, these results establish a viable route for scalable quantum information processing using donor nuclear and electron spins.
Practical quantum computers require the construction of a large network of highly coherent qubits, interconnected in a design robust against errors. Donor spins in silicon provide state-of-the-art coherence and quantum gate fidelities, in a physical platform adapted from industrial semiconductor processing. Here we present a scalable design for a silicon quantum processor that does not require precise donor placement and allows hundreds of nanometers inter-qubit distances, therefore facilitating fabrication using current technology. All qubit operations are performed via electrical means on th…
The electrical detection of spin echoes via echo tomography is used to observe decoherence processes associated with the electrical readout of the spin state of phosphorus donor electrons in silicon near a SiO$_2$ interface. Using the Carr-Purcell pulse sequence, an echo decay with a time constant of $1.7\pm0.2 \rm{μs}$ is observed, in good agreement with theoretical modeling of the interaction between donors and paramagnetic interface states. Electrical spin echo tomography thus can be used to study the spin dynamics in realistic spin qubit devices for quantum information processing.
As progress is made towards the first generation of error-corrected quantum computers, robust characterization and validation protocols are required to assess the noise environments of physical quantum processors. While standard coherence metrics and characterization protocols such as T1 and T2, process tomography, and randomized benchmarking are now ubiquitous, these techniques provide only partial information about the dynamic multi-qubit loss channels responsible for processor errors, which can be described more fully by a Lindblad operator in the master equation formalism. Here, we introdu…
We present the first 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 speci…
The ability to transport quantum information across some distance can facilitate the design and operation of a quantum processor. One-dimensional spin chains provide a compact platform to realize scalable spin transport for a solid-state quantum computer. Here, we model odd-sized donor chains in silicon under a range of experimental non-idealities, including variability of donor position within the chain. We show that the tolerance against donor placement inaccuracies is greatly improved by operating the spin chain in a mode where the electrons are confined at the Si-SiO$_2$ interface. We then…
Donor spin in silicon have achieved record values of coherence times and single-qubit gate fidelities. The next stage of development involves demonstrating high-fidelity two-qubit logic gates, where the most natural coupling is the exchange interaction. To aid the efficient design of scalable donor-based quantum processors, we model the two-electron wave function using a full configuration interaction method within a multi-valley effective mass theory. We exploit the high computational efficiency of our code to investigate the exchange interaction, valley population, and electron densities for…
Dopant atoms are ubiquitous in semiconductor technologies, providing the tailored electronic properties that underpin the modern digital information era. Harnessing the quantum nature of these atomic-scale objects represents a new and exciting technological revolution. In this article we describe the use of ion-implanted donor spins in silicon for quantum technologies. We review how to fabricate and operate single-atom spin qubits in silicon, obtaining some of the most coherent solid-state qubits, and we discuss pathways to scale up these qubits to build large quantum processors. Heavier group…
Quantum process tomography (QPT), where a quantum channel is reconstructed through the analysis of repeated quantum measurements, is an important tool for validating the operation of a quantum processor. We detail the combined use of an existing QPT approach based on tensor networks (TNs) and unsupervised learning with TN contraction path finding algorithms in order to use TNs of arbitrary topologies for reconstruction. Experiments were conducted on the 7-qubit IBM Quantum Falcon Processor ibmq_casablanca, where we demonstrate this technique by matching the topology of the tensor networks used…
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 …
The possibility of quantum computing with spins in germanium nanoscale transistors has recently attracted interest since it promises highly tuneable qubits that have encouraging coherence times. We here present the first complete theory of the orbital states of Ge donor electrons, and use it to show that Ge could have significant advantages over silicon in the implementation of a donor-based quantum processor architecture. We show that the stronger spin-orbit interaction and the larger electron donor wave functions for Ge donors allow for greater tuning of the single qubit energy than for thos…
Crosstalk is a leading source of failure in multiqubit quantum information processors. It can arise from a wide range of disparate physical phenomena, and can introduce subtle correlations in the errors experienced by a device. Several hardware characterization protocols are able to detect the presence of crosstalk, but few provide sufficient information to distinguish various crosstalk errors from one another. In this article we describe how gate set tomography, a protocol for detailed characterization of quantum operations, can be used to identify and characterize crosstalk errors in quantum…
We experimentally study the coupling of Group V donor spins in silicon to mechanical strain, and measure strain-induced frequency shifts which are linear in strain, in contrast to the quadratic dependence predicted by the valley repopulation model (VRM), and therefore orders of magnitude greater than that predicted by the VRM for small strains $|\varepsilon| < 10^{-5}$. Through both tight-binding and first principles calculations we find that these shifts arise from a linear tuning of the donor hyperfine interaction term by the hydrostatic component of strain and achieve semi-quantitative agre…
Precisely engineered mechanical oscillators keep time, filter signals, and sense motion, making them an indispensable part of today's technological landscape. These unique capabilities motivate bringing mechanical devices into the quantum domain by interfacing them with engineered quantum circuits. Proposals to combine microwave-frequency mechanical resonators with superconducting devices suggest the possibility of powerful quantum acoustic processors. Meanwhile, experiments in several mechanical systems have demonstrated quantum state control and readout, phonon number resolution, and phonon-…
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 quantu…
Quantum process tomography has become increasingly critical as the need grows for robust verification and validation of candidate quantum processors. Here, we present an approach for efficient quantum process tomography that uses a physically motivated ansatz for an unknown quantum process. Our ansatz bootstraps to an effective description for an unknown process on a multi-qubit processor from pairwise two-qubit tomographic data. Further, our approach can inherit insensitivity to system preparation and measurement error from the two-qubit tomography scheme. We benchmark our approach using nume…
Silicon spin qubits satisfy the necessary criteria for quantum information processing. However, a demonstration of high fidelity state preparation and readout combined with high fidelity single- and two-qubit gates, all of which must be present for quantum error correction, has been lacking. We use a two qubit Si/SiGe quantum processor to demonstrate state preparation and readout with fidelity over 97%, combined with both single- and two-qubit control fidelities exceeding 99%. The operation of the quantum processor is quantitatively characterized using gate set tomography and randomized benchm…
Silicon nanoelectronic devices can host single-qubit quantum logic operations with fidelity better than 99.9%. For the spins of an electron bound to a single donor atom, introduced in the silicon by ion implantation, the quantum information can be stored for nearly 1 second. However, manufacturing a scalable quantum processor with this method is considered challenging, because of the exponential sensitivity of the exchange interaction that mediates the coupling between the qubits. Here we demonstrate the conditional, coherent control of an electron spin qubit in an exchange-coupled pair of $^{…
We report the characterization of a two-qubit processor implemented with two capacitively coupled tunable superconducting qubits of the transmon type, each qubit having its own non-destructive single-shot readout. The fixed capacitive coupling yields the \sqrt{iSWAP} two-qubit gate for a suitable interaction time. We reconstruct by state tomography the coherent dynamics of the two-bit register as a function of the interaction time, observe a violation of the Bell inequality by 22 standard deviations after correcting readout errors, and measure by quantum process tomography a gate fidelity of 9…
Quantum gates in experiment are inherently prone to errors that need to be characterized before they can be corrected. Full characterization via quantum process tomography is impractical and often unnecessary. For most practical purposes, it is enough to estimate more general quantities such as the average fidelity. Here we use a unitary 2-design and twirling protocol for efficiently estimating the average fidelity of Clifford gates, to certify a 7-qubit entangling gate in a nuclear magnetic resonance quantum processor. Compared with more than $10^8$ experiments required by full process tomogr…
As a possible physical realization of a quantum information processor, a system with stacked self-assembled InAs quantum dots buried in GaAs in adjacent to the channel of a spin field-effect transistor has been proposed. In this system, only one of the stacked qubits, i.e. the edge qubit (the qubit closest to the channel), is measurable via "spin-blockade measurement." It is shown that the state tomography of the whole chain of the qubits is still possible even under such a restricted accessibility. The idea is to make use of the entangling dynamics of the qubits. A recipe for the two-qubit sy…
Quantum tomography is a widely applicable tool for complete characterization of quantum states and processes. In the present work, we develop a method for precision-guaranteed quantum process tomography. With the use of the Choi-Jamiolkowski isomorphism, we generalize the recently suggested extended norm minimization estimator for the case of quantum processes. Our estimator is based on the Hilbert-Schmidt distance for quantum processes. Specifically, we discuss the application of our method for characterizing quantum gates of a superconducting quantum processor in the framework of the IBM Q E…
Interaction of molecules with light may lead to electronic transitions and simultaneous vibrational excitations. Franck-Condon factors (FCFs) play an important role in quantifying the intensities of such vibronic transitions occurring during molecular photo-excitations. In this article, we describe a general method for estimating FCFs using a quantum information processor. The method involves the application of a translation operator followed by the measurement of certain projections. We also illustrate the method by experimentally estimating FCFs with the help of a three-qubit NMR quantum inf…
With qubit measurement and control fidelities above the threshold of fault-tolerance, much attention is moving towards the daunting task of scaling up the number of physical qubits to the large numbers needed for fault tolerant quantum computing. Here, quantum dot based spin qubits may offer significant advantages due to their potential for high densities, all-electrical operation, and integration onto an industrial platform. In this system, the initialisation, readout, single- and two-qubit gates have been demonstrated in various qubit representations. However, as seen with other small scale …
PyGSTi is a Python software package for assessing and characterizing the performance of quantum computing processors. It can be used as a standalone application, or as a library, to perform a wide variety of quantum characterization, verification, and validation (QCVV) protocols on as-built quantum processors. We outline pyGSTi's structure, and what it can do, using multiple examples. We cover its main characterization protocols with end-to-end implementations. These include gate set tomography, randomized benchmarking on one or many qubits, and several specialized techniques. We also discuss …
As increasingly impressive quantum information processors are realized in laboratories around the world, robust and reliable characterization of these devices is now more urgent than ever. These diagnostics can take many forms, but one of the most popular categories is tomography, where an underlying parameterized model is proposed for a device and inferred by experiments. Here, we introduce and implement efficient operational tomography, which uses experimental observables as these model parameters. This addresses a problem of ambiguity in representation that arises in current tomographic app…
Quantum process tomography of each directly implementable quantum gate used in the IBM quantum processors is performed to compute gate error in order to check viability of complex quantum operations in the superconductivity-based quantum computers introduced by IBM and to compare the quality of these gates with the corresponding gates implemented using other technologies. Quantum process tomography (QPT) of C-NOT gates have been performed for three configurations available in IBM QX4 processor. For all the other allowed gates QPT have been performed for every allowed position (i.e., by placing…
Random local measurements have recently been proposed to construct entanglement witnesses and thereby detect the presence of bipartite entanglement. We experimentally demonstrate the efficacy of one such scheme on a two-qubit NMR quantum information processor. We show that a set of three random local measurements suffices to detect the entanglement of a general two-qubit state. We experimentally generate states with different amounts of entanglement, and show that the scheme is able to clearly witness entanglement. We perform complete quantum state tomography for each state and compute state f…
A challenge for scaling up quantum processors using frequency-crowded, weakly anharmonic qubits is to drive individual qubits without causing leakage into non-computational levels of the others, while also minimizing the number of control lines. To address this, we implement single-qubit Wah-Wah control in a circuit QED processor with a single feedline for all transmon qubits, operating at the maximum gate speed achievable given the frequency crowding. Randomized benchmarking and quantum process tomography confirm alternating qubit control with $\leq$1% average error per computational step and…
Quantum state tomography (QST) represents an essential tool for the characterization, verification, and validation (QCVV) of quantum processors. Only for a few idealized scenarios, there are analytic results for the optimal measurement set for QST. E.g., in a setting of non-degenerate measurements, an optimal minimal set of measurement operators for QST has eigenbases which are mutually unbiased. However, in other set-ups, dependent on the rank of the projection operators and the size of the quantum system, the optimal choice of measurements for efficient QST needs to be numerically approximat…
Superconducting quantum circuits are promising candidate for building scalable quantum computers. Here, we use a four-qubit superconducting quantum processor to solve a two-dimensional system of linear equations based on a quantum algorithm proposed by Harrow, Hassidim, and Lloyd [Phys. Rev. Lett. \textbf{103}, 150502 (2009)], which promises an exponential speedup over classical algorithms under certain circumstances. We benchmark the solver with quantum inputs and outputs, and characterize it by non-trace-preserving quantum process tomography, which yields a process fidelity of $0.837\pm0.006…
In the current work we address the problem of quantum process tomography (QPT) in the case of imperfect preparation and measurement of the states which are used for QPT. The fuzzy measurements approach which helps us to efficiently take these imperfections into account is considered. However, to implement such a procedure one should have a detailed information about the errors. An approach for obtaining the partial information about them is proposed. It is based on the tomography of the ideal identity gate. This gate could be implemented by performing the measurement right after the initial st…
Medium-scale quantum devices that integrate about hundreds of physical qubits are likely to be developed in the near future. However, such devices will lack the resources for realizing quantum fault tolerance. Therefore, the main challenge of exploring the advantage of quantum computation is to minimize the impact of device and control imperfections without encoding. Quantum error mitigation is a solution satisfying the requirement. Here, we demonstrate an error mitigation protocol based on gate set tomography and quasiprobability decomposition. One- and two-qubit circuits are tested on a supe…
Error models for quantum computing processors describe their deviation from ideal behavior and predict the consequences in applications. But those processors' experimental behavior -- the observed outcome statistics of quantum circuits -- are rarely consistent with error models, even in characterization experiments like randomized benchmarking (RB) or gate set tomography (GST), where the error model was specifically extracted from the data in question. We show how to resolve these inconsistencies, and quantify the rate of unmodeled errors, by augmenting error models with a parameterized wildca…
Gates in error-prone quantum information processors are often modeled using sets of one- and two-qubit process matrices, the standard model of quantum errors. However, the results of quantum circuits on real processors often depend on additional external "context" variables. Such contexts may include the state of a spectator qubit, the time of data collection, or the temperature of control electronics. In this article we demonstrate a suite of simple, widely applicable, and statistically rigorous methods for detecting context dependence in quantum circuit experiments. They can be used on any d…
With nowadays steadily growing quantum processors, it is required to develop new quantum tomography tools that are tailored for high-dimensional systems. In this work, we describe such a computational tool, based on recent ideas from non-convex optimization. The algorithm excels in the compressed-sensing-like setting, where only a few data points are measured from a low-rank or highly-pure quantum state of a high-dimensional system. We show that the algorithm can practically be used in quantum tomography problems that are beyond the reach of convex solvers, and, moreover, is faster than other …
We undertake experimental detection of the entanglement present in arbitrary three-qubit pure quantum states on an NMR quantum information processor. Measurements of only four observables suffice to experimentally differentiate between the six classes of states which are inequivalent under stochastic local operation and classical communication (SLOCC). The experimental realization is achieved by mapping the desired observables onto Pauli $z$-operators of a single qubit, which is directly amenable to measurement. The detection scheme is applied to known entangled states as well as to states ran…
We experimentally demonstrate that loop state-preparation-and-measurement (SPAM) tomography is capable of detecting correlated errors in a two-qubit system. We prepare photon pairs in a state that approximates a Werner state, which may or may not be entangled. By performing measurements with multiple different detector settings we are able to detect correlated errors between two single-qubit measurements performed in different locations. No assumptions are made concerning either the state preparations or the measurements, other than that the dimensions of the states and the positive-operator-v…
We present an experimental implementation of the coined discrete time quantum walk on a square using a three qubit liquid state nuclear magnetic resonance (NMR) quantum information processor (QIP). Contrary to its classical counterpart, we observe complete interference after certain steps and a periodicity in the evolution. Complete state tomography has been performed for each of the eight steps making a full period. The results have extremely high fidelity with the expected states and show clearly the effects of quantum interference in the walk. We also show and discuss the importance of choo…
