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Quantum error correction represents a significant milestone in large-scale quantum computing, with the surface code being a prominent strategy due to its high error threshold and experimental feasibility. However, it is challenging to implement non-Clifford logical gates in a fault-tolerant way with low overhead, through the conventional magic state distillation technique. Here, we enhance the performance of the conventional surface code by incorporating non-Pauli stabilizers and introduce two innovative initialization protocols. Our approach enhances the fidelity of the initialization of non-Clifford logical state by avoiding unprotected operations before the encoding process. This improved fidelity of the initialization of non-Clifford logical states reduces the necessary number of logical qubits for precise state distillation, ultimately decreasing the overall resource overhead. Furthermore, we demonstrate the ability to entangle logical qubits in non-Pauli and Pauli bases via the lattice surgery technique. This integration enables the use of Pauli-based surface codes for computation while non-Pauli codes are employed for auxiliary qubit initialization, thus compatible with the conventional wisdom of logical Clifford operation based on the Pauli-based surface code.

A quantum state for being an eigenstate of some local Hamiltonian should be constraint by zero energy variance and consequently, the constraint is rather strong that a single eigenstate may uniquely determine the Hamiltonian. For non-Hermitian systems, it is natural to expect that determining the Hamiltonian requires a pair of both left and right eigenstates. Here, we observe that it can be sufficient to determine a non-Hermitian Hamiltonian from a single right or left eigenstate. Our approach is based on the quantum covariance matrix, where the solution of Hamiltonian corresponds to the complex null vector. Our scheme favours non-Hermitian Hamiltonian learning on experimental quantum systems, as only the right eigenstates there can be accessed. Furthermore, we use numerical simulations to examine the effects of measurement errors and show the stability of our scheme.

High-fidelity quantum gates are an essential prerequisite for large-scale quantum computation. When manipulating practical quantum systems, environmentally and operationally induced errors are inevitable, and thus, in addition to being fast, it is preferable that operations should be intrinsically robust against different errors. Here, we propose a general protocol for constructing geometric quantum gates with on-demand trajectories by modulating the applied pulse shapes that define the system's evolution trajectory. Our scheme adopts reverse engineering of the target Hamiltonian using smooth pulses, which also eliminates the difficulty of calculating geometric phases for an arbitrary trajectory. Furthermore, because a particular geometric gate can be induced by various different trajectories, we can further optimize the gate performance under different scenarios; the results of numerical simulations indicate that this optimization can greatly enhance the quality of the gate. In addition, we present an implementation of our proposal using superconducting circuits, showcasing substantial enhancements in gate performance compared with conventional schemes. Our protocol thus presents a promising approach for high-fidelity and strong-robust geometric quantum gates for large-scale quantum computation.

Geometric phases are robust to local noises and the nonadiabatic ones can reduce the evolution time, thus nonadiabatic geometric gates have strong robustness and can approach high fidelity. However, the advantage of geometric phase has not being fully explored in previous investigations. Here, we propose a scheme for universal quantum gates with pure nonadiabatic and noncyclic geometric phases from smooth evolution paths. In our scheme, only geometric phase can be accumulated in a fast way, and thus it not only fully utilizes the local noise resistant property of geometric phase but also reduces the difficulty in experimental realization. Numerical results show that the implemented geometric gates have stronger robustness than dynamical gates and the geometric scheme with cyclic path. Furthermore, we propose to construct universal quantum gate on superconducting circuits, with the fidelities of single-qubit gate and nontrivial two-qubit gate can achieve $99.97\%$ and $99.87\%$, respectively. Therefore, these high-fidelity quantum gates are promising for large-scale fault-tolerant quantum computation.

High-fidelity universal quantum gates are widely acknowledged as essential for scalable quantum computation. However, in solid-state quantum systems, which hold promise as physical implementation platforms for quantum computation, the inevitable $ZZ$ crosstalk resulting from interqubit interactions significantly impairs quantum operation performance. Here we propose a scalable protocol to achieve $ZZ$-crosstalk mitigation in universal quantum gates. This method converts the noisy Hamiltonian with $ZZ$ crosstalk into a framework that efficiently suppresses all $ZZ$-crosstalk effects, leading to ideal target quantum operations. Specifically, we first analytically derive the $ZZ$-crosstalk mitigation conditions and then apply them to enhance the performance of target universal quantum gates. Moreover, numerical simulations validate the effectiveness of $ZZ$-crosstalk mitigation when multiple qubit gates operate concurrently. As a result, our protocol presents a promising approach for implementing practical parallel quantum gates in large-scale quantum computation scenarios.

Geometric phase has the intrinsic property of being resistant to some types of local noises as it only depends on global properties of the evolution path. Meanwhile, the non-Abelian geometric phase is in the matrix form, and thus can naturally be used to implement high performance quantum gates, i.e., the so-called holonomic quantum computation. This article reviews recent advances in nonadiabatic holonomic quantum computation, and focuses on various optimal control approaches that can improve the gate performance, in terms of the gate fidelity and robustness. Besides, we also pay special attention to its possible physical realizations and some concrete examples of experimental realizations. Finally, with all these efforts, within state-of-the-art technology, the performance of the implemented holonomic quantum gates can outperform the conventional dynamical ones, under certain conditions.

Universal robust quantum control is essential for performing complex quantum algorithms and efficient quantum error correction protocols. Geometric phase, as a key element with intrinsic fault-tolerant feature, can be well integrated into quantum control processes to enhance control robustness. However, the current geometric quantum control is still controversial in robust universality, which leads to the unsatisfactory result that cannot sufficiently enhance the robustness of arbitrary type of geometric gate. In this study, we find that the finite choice on geometric evolution trajectory is one of the main roots that constrain the control robustness of previous geometric schemes, as it is unable to optionally avoid some trajectory segments that are seriously affected by systematic errors. In view of this, we here propose a new scheme for universal robust geometric control based on geometric trajectory correction, where enough available evolution parameters are introduced to ensure that the effective correction against systematic errors can be executed. From the results of our numerical simulation, arbitrary type of geometric gate implemented by using the corrected geometric trajectory has absolute robustness advantages over conventional quantum one. In addition, we also verify the feasibility of the high-fidelity physical implementation of our scheme in superconducting quantum circuit, and finally discuss in detail the potential researches based on our scheme. Therefore, our theoretical work is expected to offer an attractive avenue for realizing practical fault-tolerant quantum computation in existing experimental platforms., Comment: 10 pages, 5 figures

Solving non-Hermitian quantum many-body systems on a quantum computer by minimizing the variational energy is challenging as the energy can be complex. Here, based on energy variance, we propose a variational method for solving the non-Hermitian Hamiltonian, as zero variance can naturally determine the eigenvalues and the associated left and right eigenstates. Moreover, the energy is set as a parameter in the cost function and can be tuned to obtain the whole spectrum, where each eigenstate can be efficiently obtained using a two-step optimization scheme. Through numerical simulations, we demonstrate the algorithm for preparing the left and right eigenstates, verifying the biorthogonal relations, as well as evaluating the observables. We also investigate the impact of quantum noise on our algorithm and show that its performance can be largely improved using error mitigation techniques. Therefore, our work suggests an avenue for solving non-Hermitian quantum many-body systems with variational quantum algorithms on near-term noisy quantum computers.

The nonadiabatic holonomic quantum computation based on three-level systems has wide applicability experimentally due to its simpler energy level structure requirement and inherent robustness from the geometric phase. However, in previous conventional schemes, the states of the calculation subspace have always leaked to the noncomputation subspace, resulting in less robustness than anticipated. Recent efforts to address this problem are at the cost of excessively long gate time, which will lead to more decoherence-induced errors. Here, we propose a solution to the problem without the severe limitation of the much longer gate time. Specifically, we implement arbitrary holonomic gates via a three-segment Hamiltonian, where the gate time depends on the rotation angle, and the smaller the rotation angle, the shorter the gate time will be. Compared with the previous solutions, our numerical simulations indicate that the decoherence-induced gate errors of our scheme are greatly decreased and the robustness of our scheme is also better, particularly for small-angle rotation gates. Moreover, we provide a detailed physical realization of our proposal on a two-dimensional superconducting quantum circuit. Therefore, our protocol provides a promising alternative for future fault-tolerant quantum computation., Comment: accepted version

Recently, nonadiabatic geometric quantum computation has been received great attentions, due to its fast operation and intrinsic error resilience. However, compared with the corresponding dynamical gates, the robustness of implemented nonadiabatic geometric gates based on the conventional single-loop scheme still has the same order of magnitude due to the requirement of strict multi-segment geometric controls, and the inherent geometric fault-tolerance characteristic is not fully explored. Here, we present an effective geometric scheme combined with a general dynamical-corrected technique, with which the super-robust nonadiabatic geometric quantum gates can be constructed over the conventional single-loop and two-loop composite-pulse strategies, in terms of resisting the systematic error, i.e., $\sigma_x$ error. In addition, combined with the decoherence-free subspace (DFS) coding, the resulting geometric gates can also effectively suppress the $\sigma_z$ error caused by the collective dephasing. Notably, our protocol is a general one with simple experimental setups, which can be potentially implemented in different quantum systems, such as Rydberg atoms, trapped ions and superconducting qubits. These results indicate that our scheme represents a promising way to explore large-scale fault-tolerant quantum computation., Comment: 10 pages, 9 figures

Decoherence is inevitable when manipulating quantum systems. It decreases the quality of quantum manipulations and thus is one of the main obstacles for large-scale quantum computation, where high-fidelity quantum gates are needed. Generally, the longer a gate operation is, the more decoherence-induced gate infidelity will be. Therefore, how to shorten the gate time becomes an urgent problem to be solved. To this end, time-optimal control based on solving the quantum brachistochrone equation is a straightforward solution. Here, based on time-optimal control, we propose a scheme to realize universal quantum gates on superconducting qubits in a two-dimensional square lattice configuration, and the two-qubit gate fidelity approaches 99.9\%. Meanwhile, we can further accelerate the Z-axis gate considerably by adjusting the detuning of the external driving. Finally, in order to reduce the influence of the dephasing error, decoherence-free subspace encoding is also incorporated in our physical implementation. Therefore, we present a fast quantum scheme which is promising for large-scale quantum computation., Comment: v2 accepted

Quantum computation has demonstrated advantages over classical computation for special hard problems, where a set of universal quantum gates is essential. Geometric phases, which have built-in resilience to local noise, have been used to construct quantum gates with excellent performance. However, this advantage has been smeared in previous schemes. Here, we propose a state-independent nonadiabatic geometric quantum-gate scheme that is able to realize a more fully geometric gate than previous approaches, allowing for the cancelation of dynamical phases accumulated by an arbitrary state. Numerical simulations demonstrate that our scheme has significantly stronger gate robustness than the previous geometric and dynamical ones. Meanwhile, we give a detailed physical implementation of our scheme with the Rydberg atom system based on the Rydberg blockade effect, specifically for multiqubit control-phase gates, which exceeds the fault-tolerance threshold of multiqubit quantum gates within the considered error range. Therefore, our scheme provides a promising way for fault-tolerant quantum computation in atomic systems.

Quantum computation attaches importance to high-precision quantum manipulation, where the quantum state transfer with high fidelity is necessary. Here, we propose a new scheme to implement the quantum state transfer of high fidelity and long distance, by adding on-site potential into the qubit chain and enlarging the proportion of the coupling strength between the two ends and the chain. In the numerical simulation, without decoherence, the transfer fidelities of 9 and 11 qubit chain are 0.999 and 0.997, respectively. Moreover, we give a detailed physical realization scheme of the quantum state transfer in superconducting circuits, and discuss the tolerance of our proposal against decoherence. Therefore, our scheme will shed light on quantum computation with long chain and high-fidelity quantum state transfer., Comment: 5 pages, 4 figures

It is well-known that time-dependent Schr\"{o}dinger equation can only be exactly solvable in very rare cases, even for two-level quantum systems. Therefore, finding exact quantum dynamics under time-dependent Hamiltonian is not only of fundamental importance in quantum physics but also can facilitate active quantum manipulations for quantum information processing. Here, we present a method which could generate a near infinite number of analytical-assisted solutions of the Schr\"{o}dinger equation for a qubit with time-dependent driving. This analytical-assisted solution has free parameters with only boundary restrictions, and thus can find many applications in precise quantum manipulations. Due to the general form of the time-dependent Hamiltonian in our scheme, it can be readily implemented in various experimental setups of qubits. Therefore, our scheme provides new solutions for Schr\"{o}dinger equation, thus provides an alternative and analytical-based routine for precise control over qubits.

Quantum gates based on geometric phases possess intrinsic noise-resilience features and therefore attract much attention. However, the implementations of previous geometric quantum computation typically require a long pulse time of gates. As a result, their experimental control inevitably suffers from the cumulative disturbances of systematic errors due to excessive time consumption. Here, we experimentally implement a set of noncyclic and nonadiabatic geometric quantum gates in a superconducting circuit, which greatly shortens the gate time. And also, we experimentally verify that our universal single-qubit geometric gates are more robust to both the Rabi frequency error and qubit frequency shift-induced error, compared to the conventional dynamical gates, by using the randomized benchmarking method. Moreover, this scheme can be utilized to construct two-qubit geometric operations, while the generation of the maximally entangled Bell states is demonstrated. Therefore, our results provide a promising routine to achieve fast, high-fidelity, and error-resilient quantum gates in superconducting quantum circuits.

We investigate the scalar perturbations in a class of spatially covariant gravity theory with a dynamical lapse function. Generally, there are two scalar degrees of freedom due to the presence of the velocity of the lapse function. We treat the scalar perturbations as analogues of those in a two-field inflationary mode, in which one is light mode and the other is the heavy mode. This is justified by the fact that the scalar mode due to the dynamical lapse function becomes infinitely heavy in the limit when the lapse function reduces to be an auxiliary variable. The standard approaches of multiple filed perturbations can be applied to deal with our model. By integrating out the heavy mode and derive the effective theory for the single light field, we find the solution to the single mode in the form of plane waves. Then we calculate the corrections to the power spectrum of the light mode from the heavy mode, by making use of the standard perturbative method of field theory. At last, when the two fields are not weakly coupled, we find a power law mode for the coupled system in large scales.

Besides the intrinsic noise resilience property, nonadiabatic geometric phases are of the fast evolution nature, and thus can naturally be used in constructing quantum gates with excellent performance, i.e., the so-called nonadiabatic geometric quantum computation (NGQC). However, previous single-loop NGQC schemes are sensitive to the operational control error, i.e., the $X$ error, due to the limitations of the implementation. Here, we propose a robust scheme for NGQC combining with the dynamical correction technique, which still uses only simplified pulses, and thus being experimental friendly. We numerically show that our scheme can greatly improve the gate robustness in previous protocols, retaining the intrinsic merit of geometric phases. Furthermore, to fight against the dephasing noise, due to the $Z$ error, we can incorporate the decoherence-free subspace encoding strategy. In this way, our scheme can be robust against both types of errors. Finally, we also propose how to implement the scheme with encoding on superconducting quantum circuits with experimentally demonstrated technology. Therefore, due to the intrinsic robustness, our scheme provides a promising alternation for the future scalable fault-tolerant quantum computation., Comment: 6 pages, 5 figures

Confinement of quarks due to the strong interaction and the deconfinement at high temperatures and high densities are a basic paradigm for understanding the nuclear matter. Their simulation, however, is very challenging for classical computers due to the sign problem of solving equilibrium states of finite-temperature quantum chromodynamical systems at finite density. In this paper, we propose a variational approach, using the lattice Schwinger model, to simulate the confinement or deconfinement by investigating the string tension. We adopt an ansatz that the string tension can be evaluated without referring to quantum protocols for measuring the entropy in the free energy. Results of numeral simulation show that the string tension decreases both along the increasing of the temperature and the chemical potential, which can be an analog of the phase diagram of QCD. Our work paves a way for exploiting near-term quantum computers for investigating the phase diagram of finite-temperature and finite density for nuclear matters.

Nonadiabatic holonomic quantum computation (NHQC) is implemented by fast evolution processes in a geometric way to withstand local noises. However, recent works of implementing NHQC are sensitive to the systematic noise and error. Here, we present a path-optimized NHQC (PONHQC) scheme based on the non-Abelian geometric phase, and find that a geometric gate can be constructed by different evolution paths, which have different responses to systematic noises. Due to the flexibility of the PONHQC scheme, we can choose an optimized path that can lead to excellent gate performance. Numerical simulation shows that our optimized scheme can greatly outperform the conventional NHQC scheme, in terms of both fidelity and robustness of the gates. In addition, we propose to implement our strategy on superconducting quantum circuits with decoherence-free subspace encoding with the experiment-friendly two-body exchange interaction. Therefore, we present a flexible NHQC scheme that is promising for the future robust quantum computation., Comment: 9 pages, 4 figures

Quantum state transfer (QST) and entangled state generation (ESG) are important building blocks for modern quantum information processing. To achieve these tasks, convention wisdom is to consult the quantum adiabatic evolution, which is time-consuming, and thus is of low fidelity. Here, using the shortcut to adiabaticity technique, we propose a general method to realize high-fidelity fast QST and ESG in a cavity-coupled many qubits system via its dark pathways, which can be further designed for high-fidelity quantum tasks with different optimization purpose. Specifically, with a proper dark pathway, QST and ESG between any two qubits can be achieved without decoupling the others, which simplifies experimental demonstrations. Meanwhile, ESG among all qubits can also be realized in a single step. In addition, our scheme can be implemented in many quantum systems, and we illustrate its implementation on superconducting quantum circuits. Therefore, we propose a powerful strategy for selective quantum manipulation, which is promising in cavity coupled quantum systems and could find many convenient applications in quantum information processing., Comment: Accepted for publication in Frontiers of Physics

Along with the scaling of dimensions in quantum systems, transitions between the system's energy levels would become close in frequency, which are conventionally resolved by weak and lengthy pulses. Here, we extend and experimentally demonstrate analytically based swift quantum control techniques on a four-level trapped ion system, where we perform individual or simultaneous control over two pairs of spectrally nearby transitions with tailored time-varied drive, achieving operational fidelities ranging from 99.2(3)\% to 99.6(3)\%. Remarkably, we achieve approximately an order of magnitude speed up comparing with the case of weak square pulse for a general control. Therefore, our demonstration may be beneficial to a broad range of quantum systems with crowded spectrum, for spectroscopy, quantum information processing and quantum simulation., Comment: 9 pages, 5 figures

Geometric quantum computation offers a practical strategy toward robust quantum computation due to its inherently error tolerance. However, the rigorous geometric conditions lead to complex and/or error-disturbed quantum controls, especially for logical qubits that involve more physical qubits, whose error tolerance is effective in principle though, their experimental demonstration is still demanding. Thus, how to best simplify the needed control and manifest its full advantage has become the key to widespread applications of geometric quantum computation. Here we propose a new fast and robust geometric scheme, with the decoherence-free-subspace encoding, and present its physical implementation on superconducting quantum circuits, where we only utilize the experimentally demonstrated parametrically tunable coupling to achieve high-fidelity geometric control over logical qubits. Numerical simulation verifies that it can efficiently combine the error tolerance from both the geometric phase and logical-qubit encoding, displaying our gate-performance superiority over the conventional dynamical one without encoding, in terms of both gate fidelity and robustness. Therefore, our scheme can consolidate both error suppression methods for logical-qubit control, which sheds light on the future large-scale quantum computation., Comment: 8 pages, 5 figures