Measuring two-qubit gates

We introduce two methods for quantum process and detector tomography. In the quantum process tomography method, we develop an analytical procedure for projecting the linear inversion estimation of a quantum channel onto the set of completely positive trace-preserving matrices. By integrating this method with alternate projection techniques, we achieve a three-order-of-magnitude improvement in approximating the closest quantum channel to an arbitrary Hermitian matrix compared to existing methods without compromising computational efficiency. Our second method extends this approach to quantum detector tomography, demonstrating superior efficiency compared to current techniques. Through numerical simulations, we evaluate our protocols across channels of up to four qubits in quantum process tomography and systems of up to six qubits in quantum detector tomography, showcasing superior precision and efficiency.

Quantum device characterization via state tomography plays an important role in both validating quantum hardware and processing quantum information, but it needs the exponential number of the measurements. For the systems with XX+YY-type couplings and Z readouts, such as superconducting quantum computing (SQC) systems, traditional quantum state tomography (QST) using single-qubit readout operations at least requires $3^n$ measurement settings in reconstructing an $n$-qubit state. In this work, I proposed an improved QST by adding 2-qubit evolutions as the readout operations and obtained an optimal tomographic scheme using the integer programming optimization. I respectively apply the new scheme on SQC systems with the Nearest-Neighbor, 2-Dimensional, and All-to-All connectivities on qubits. It shows that this method can reduce the number of measurements by over 60% compared with the traditional QST. Besides, comparison with the traditional scheme in the experimental feasibility and robustness against errors were made by numerical simulation. It is found that, the new scheme has good implementability and it can achieve comparable or even better accuracy than the traditional scheme. It is expected that the experimentalist from the related fields can directly utilize the ready-made results for reconstructing quantum states involved in their research.

A general approach to the provision of the quality and efficiency of quantum information technologies is proposed. The approach is based on the analysis of quantum noise generated on the implementation of quantum operations. The method of precision quantum measurements of logic quantum gates based on superconducting phase qubits is developed. The universal method of quantum state and process tomography is used to carry out a comprehensive analysis of the fidelity of the tomography of two-qubit SQiSW, CNOT, and CZ gates arising from the capacitive coupling between qubits. The method of optimization of quantum process tomography is proposed to provide a substantially higher degree of adequacy and fidelity compared to those attained by previously known methods. The examples of simulation of depolarizing quantum noise and amplitude- and phase-relaxation processes on the implementation of quantum operations in registers are considered.

In NMR quantum computing, spin states of spin-1/2 nuclei are called qubits. Quantum logic gates are represented by unitary matrices. As a universal gate, controlled-NOT (CNOT) is a two-qubit gate. For the IS (I = 1/2 and S = 1/2) spin system, two-qubit CNOT gate is represented by a 4 × 4 matrix. SWAP logic gate, which exchanges two quantum states, is constructed by CNOT gates. In this study, first, four-qubit CNOT gates are constructed for the IS (I = 3/2, S = 3/2) spin system. Then, by using these CNOT gates, a four-qubit SWAP logic gate is found. As an application and verification, an obtained SWAP logic gate is applied to the matrix representation of product operators for the IS (I = 3/2, S = 3/2) spin system. SWAP logic gate can also be presented by an NMR pulse sequence. By using the product operator theory, the pulse sequence of the SWAP logic gate is applied to product operators of the IS (I = 3/2, S = 3/2) spin system. The expected exchange results are obtained for both the matrix representation and the pulse sequence of SWAP logic gate.

Besides noticeable challenges in implementing low-error single- and two-qubit quantum gates in superconducting quantum processors, the readout technique and analysis are a key factor in determining the efficiency and performance of quantum processors. Being able to efficiently implement quantum algorithms involving entangling gates and asses their output is mandatory for quantum utility. In a transmon-based 5-qubit superconducting quantum processor, we compared the performance of quantum circuits involving an increasing level of complexity, from single-qubit circuits to maximally entangled Bell circuits. This comparison highlighted the importance of the readout analysis and helped us optimize the protocol for more advanced quantum algorithms. Here, we report the results obtained from the analysis of the outputs of quantum circuits using two readout paradigms, referred to as “multiplied readout probabilities” and “conditional readout probabilities.” The first method is suitable for single-qubit circuits, while the second is essential for accurately interpreting the outputs of circuits involving two-qubit gates.

Nonlocal advantage of quantum coherence (NAQC) based on coherence complementarity relations is generally viewed as a stronger nonclassical correlation than Bell nonlocality. An arbitrary two-qubit state with NAQC must be an entangled state, which demonstrates that the criterion of NAQC can also be regarded as an entanglement witness. In this paper, we experimentally investigate the NAQC for Bell-diagonal states with high fidelity in an optics-based platform. We perform local measurements on a subsystem in three mutually unbiased bases and reconstruct the density matrices of the measured states by quantum state tomography process. By analyzing characteristic of the $l_1$ norm, relative entropy and skew information of coherence with parameters of quantum states, NAQC for the quantum states is accurately captured, and it shows that our experimental results are well compatible with the theoretical predictions. It is worth mentioning that quantum states with NAQC would have higher entanglement, and thus NAQC could be expected to be a kind of useful physical resource for quantum information processing.

In this work we develop new methods for the characterisation of continuous variable quantum states using heterodyne measurement in both the trusted and untrusted settings. First, building on quantum state tomography with heterodyne detection, we introduce a reliable method for continuous variable quantum state certification, which directly yields the elements of the density matrix of the state considered with analytical confidence intervals. This method neither needs mathematical reconstruction of the data nor discrete binning of the sample space, and uses a single Gaussian measurement setting. Second, beyond quantum state tomography and without its identical copies assumption, we promote our reliable tomography method to a general efficient protocol for verifying continuous variable pure quantum states with Gaussian measurements against fully malicious adversaries, i.e., making no assumptions whatsoever on the state generated by the adversary. These results are obtained using a new analytical estimator for the expected value of any operator acting on a continuous variable quantum state with bounded support over the Fock basis, computed with samples from heterodyne detection of the state.

Two-qubit entangling gates for superconducting quantum computers

Quantum state engineering and state characterization is a key task in quantum information processing in both discrete and continuous variable systems in the optical domain. In particular, quantum states with non-Gaussian (i.e., non-positive) Wigner quasiprobability distribution functions are crucial to universal, fault-tolerant quantum computing with continuous variables. In this talk, we present our recent results on single-photon Fock state tomography using Photon-Number-Resolving (PNR) measurements. We generated a highly pure narrow-band single-photon Fock state by heralding cavity-enhanced spontaneous-parametric-downconversion from a PPKTP optical parametric oscillator. The Wigner function was reconstructed with photon statistics obtained using superconducting transition-edge sensors with an overall system efficiency of 58(2)%. We then discuss quantum state engineering for pure displaced single-photon Fock states, optical cat states, and approximate GKP states using coherent states and single-photon states along with linear optics and PNR measurements. We report our experimental progress for the same.

4 - Introduction to NMR quantum computing

As quantum devices make steady progress towards intermediate scale and fault-tolerant quantum computing, it is essential to develop rigorous and efficient measurement protocols that account for known sources of noise. Most existing quantum characterization protocols such as gate-set tomography and randomized benchmarking assume the noise acting on the qubits is Markovian. However, this assumption is often not valid, as for the case of 1/f charge noise or hyperfine nuclear spin noise. Here, we present a general framework for quantum process tomography (QPT) in the presence of time-correlated noise. We further introduce fidelity benchmarks that quantify the relative strength of different sources of Markovian and non-Markovian noise. As an application of our method, we perform a comparative theoretical and experimental analysis of silicon spin qubits. We first develop a detailed noise model that accounts for the dominant sources of noise and validate the model against experimental data. Applying our framework for time-correlated QPT, we find that the number of independent parameters needed to characterize one- and two-qubit gates can be compressed by 10× and 100×, respectively, when compared to the fully generic case. These compressions reduce the amount of tomographic measurements needed in experiment, while also significantly speeding up numerical simulations of noisy quantum circuit dynamics compared to time-dependent Hamiltonian simulation. Using this compressed noise model, we find good agreement between our theoretically predicted process fidelities and two-qubit interleaved randomized benchmarking fidelities of 99.8% measured in recent experiments on silicon spin qubits. More broadly, our formalism can be directly extended to develop efficient and scalable tuning protocols for high-fidelity control of large arrays of quantum devices with non-Markovian noise. Published by the American Physical Society 2024

Quantum processors rely on classical electronic controllers to manipulate and read out the quantum state. As the performance of the quantum processor improves, non-idealities in the classical controller can become the performance bottleneck for the whole quantum computer. To prevent such limitation, this paper presents a systematic study of the impact of the classical electrical signals on the qubit fidelity. All operations, i.e. single-qubit rotations, two-qubit gates and read-out, are considered, in the presence of errors in the control electronics, such as static, dynamic, systematic and random errors. Although the presented study could be extended to any qubit technology, it currently focuses on single-electron spin qubits, because of several advantages, such as purely electrical control and long coherence times, and for their potential for large-scale integration. As a result of this study, detailed electrical specifications for the classical control electronics for a given qubit fidelity can be derived, as demonstrated with specific case studies. We also discuss the effect on qubit fidelity of the performance of the general-purpose room-temperature equipment that is typically employed to control the few qubits available today. Ultimately, we show that tailor-made electronic controllers can achieve significantly lower power, cost and size, as required to support the scaling up of quantum computers.

The technique of stimulated Raman adiabatic passage (STIRAP), which allows\nefficient and selective population transfer between quantum states without\nsuffering loss due to spontaneous emission, was introduced in 1990 (Gaubatz\n\\emph{et al.}, J. Chem. Phys. \\textbf{92}, 5363, 1990). Since then STIRAP has\nemerged as an enabling methodology with widespread successful applications in\nmany fields of physics, chemistry and beyond. This article reviews the many\napplications of STIRAP emphasizing the developments since 2000, the time when\nthe last major review on the topic was written (Vitanov \\emph{et al.}, Adv. At.\nMol. Opt. Phys. \\textbf{46}, 55, 2001). A brief introduction into the theory of\nSTIRAP and the early applications for population transfer within three-level\nsystems is followed by the discussion of several extensions to multi-level\nsystems, including multistate chains and tripod systems. The main emphasis is\non the wide range of applications in atomic and molecular physics (including\natom optics, cavity quantum electrodynamics, formation of ultracold molecules,\nprecision experiments, etc.), quantum information (including single- and\ntwo-qubit gates, entangled-state preparation, etc.), solid-state physics\n(including processes in doped crystals, nitrogen-vacancy centers,\nsuperconducting circuits, etc.), and even some applications in classical\nphysics (including waveguide optics, frequency conversion, polarization optics,\netc.). Promising new prospects for STIRAP are also presented (including\nprocesses in optomechanics, detection of parity violation in molecules,\nspectroscopy of core-nonpenetrating Rydberg states, and population transfer\nwith X-ray pulses).\n

Quantum state reconstruction, or state reconstruction for short, aims at identifying an unknown quantum state ( states in quantum mechanics) on the basis of experimentally accessible data. The Quantum Optics community usually refers to this inverse problem as quantum (state) tomography while the expression quantum state estimation is often used in the field of Quantum computation. Reconstruction procedures depend on the physical context defined by the system carrying the unknown state, the experimentally accessible observables, the size of the ensemble of systems prepared in the unknown state, and the precision of the measured data. A two-level system (such as a spin-1/2, a qubit, or the two polarizations of a photon) prepared in a state with density matrix ρ is sufficient to illustrate the idea of state reconstruction. With two non-negative eigenvalues summing to one, the density matrix is a positive operator, and it depends on three real parameters. In the Bloch representation, the parameters combine to a real vector n with length |n| 1,

In recent years, the great progress of studying the quantum entanglement has been made. In the field of optics, the great success has been achieved in quantum entanglement theory and technology. Then researchers concentrate on the microwave frequency band whose frequency is lower than that of optical frequency band. The signal in the microwave frequency band has a longer wavelength, and it has the diffraction capability that the optical signal does not possess. Furthermore, it can spread further in complex environments. Now it is possible to experimentally produce squeezed state of microwave signals and spatially separated path-entangled microwave signals. It is an important issue to judge whether the microwave signals received through dual paths are in entanglement state. In this paper, we firstly introduce the method of using squeezed state of microwave and microwave beam splitter to prepare path-entangled microwave signals. Then we use entanglement witness method to detect entanglement. Through constructing the entanglement witness operator in path-entangled microwave signals, the entanglement of path-entangled microwave signals can be effectively detected. We decompose the expression of the continuous variables path-entangled microwave signals into a large number of 2 2 entangled superposition states, deduce an entangled witness operator of path-entangled microwave signals based on the principle of partial transpose criterion and entanglement witnessing, and prove that the entangled witness can be used to detect the path-entangled microwave signals. Finally, we propose a physical verification of path-entangled microwave signal entanglement. The verification can be realized as follows:firstly, we reverse the phase of a received quantum-state microwave signal by utilizing continuous variable controlled phase gate in a range of 0-, then we send two microwave signals into the two input ports of the microwave beam splitter, and we operate coincidence counting of microwave photons on the two output ports after entanglement microwave signals have passed through the microwave splitter. By analyzing the results of the whole process, we have the following conclusions:if the coincidence rate of two input signals is higher than that of non-entangled microwave signals under the same power, signals can be counted as entanglement. The proposed method can detect the entangled microwave signals more efficiently than the conventional methods, such as quantum state reconstruction, and thus reduce the detection and computational complexity. The entanglement of the two microwave quantum state signals can be observed directly by using this method. This paper provides a new idea for detecting the path-entangled microwave signals.