We design efficient controlled-rotation gates with arbitrary angle acting on three-spin encoded qubits for exchange-only quantum computation. Two pulse sequence constructions are given. The first is motivated by an analytic derivation of the efficient Fong-Wandzura sequence for an exact CNOT gate. This derivation, briefly reviewed here, is based on elevating short sequences of SWAP pulses to an entangling two-qubit gate. To go beyond CNOT, we apply a similar elevation to a modified short sequence consisting of SWAPs and one pulse of arbitrary duration. This results in two-qubit sequences that carry out controlled-rotation gates of arbitrary angle. The second construction streamlines a class of arbitrary CPHASE gates established earlier. Both constructions are based on building two-qubit sequences out of subsequences with special properties that render each step of the construction analytically tractable.
Recent experiments have demonstrated two-qubit fidelities above 99%, however, theoretically, the fidelity of CNOT operations is limited by off-resonant driving described by off-diagonal terms in the system Hamiltonian. Here we investigate these off-diagonal contributions and we propose a fidelity improvement of several orders of magnitude by using asymmetric driving. Therefore, we provide a description of an ac virtual gate based on a simple capacitance model which not only enables a high fidelity CNOT but also crosstalk reduction when scaling up spin qubit devices to larger arrays.
Single qubit rotations and two-qubit CNOT operations are crucial ingredients for universal quantum computing. While high fidelity single qubit operations have been achieved using the electron spin degree of freedom, realizing a robust CNOT gate has been a major challenge due to rapid nuclear spin dephasing and charge noise. We demonstrate an efficient resonantly-driven CNOT gate for electron spins in silicon. Our platform achieves single-qubit rotations with fidelities >99%, as verified by randomized benchmarking. Gate control of the exchange coupling allows a quantum CNOT gate to be implement…
We investigate capacitively coupled two-qubit quantum gates based on quantum dots. For exchange-only coded qubits electron spin $S$ and its projection $S_z$ are exact quantum numbers. Capacitive coupling between qubits, as distinct from interqubit exchange, preserves these quantum numbers. We prove, both analytically and numerically, that conservation of the spins of individual qubits has dramatic effect on performance of two-qubit gates. By varying the level splittings of individual qubits, $J_a$ and $J_b$, and the interqubit coupling time $t$, we can find an infinite number of triples $(J_a,…
One of the challenges when scaling up semiconductor-based quantum processors consists in the presence of crosstalk errors caused by control operations on neighboring qubits. In previous work, crosstalk in spin qubit arrays has been investigated for non-driven single qubits near individually driven quantum gates and for two simultaneously driven single-qubit gates. Nevertheless, simultaneous gates are not restricted to single-qubit operations but also include frequently used two-qubit gates such as the CNOT gate. We analyse the impact of crosstalk drives on qubit operations, such as the CNOT an…
A critical element towards the realization of scalable quantum processors is non-local coupling between nodes. Scaling connectivity beyond nearest-neighbor interactions requires the implementation of a mediating interaction often termed a 'quantum bus'. Cavity photons have long been used as a bus by the superconducting qubit community, but it has only recently been demonstrated that spin-based qubits in double quantum dot architectures can reach the strong coupling regime and exhibit spin-spin interactions via the exchange of real or virtual photons. Two-qubit gate operations are predicted in …
The CNOT gate is a two-qubit gate which is essential for universal quantum computation. A well-established approach to implement it within Majorana-based qubits relies on subsequent measurement of (joint) Majorana parities. We propose an alternative scheme which operates a protected CNOT gate via the holonomic control of a handful of system parameters, without requiring any measurement. We show how the adiabatic tuning of pair-wise couplings between Majoranas can robustly lead to the full entanglement of two qubits, insensitive with respect to small variations in the control of the parameters.
Fast operations, an easily tunable Hamiltonian, and a straightforward two-qubit interaction make charge qubits a useful tool for benchmarking device performance and exploring two-qubit dynamics. Here, we tune a linear chain of four Si/SiGe quantum dots to host two double dot charge qubits. Using the capacitance between the double dots to mediate a strong two-qubit interaction, we simultaneously drive coherent transitions to generate correlations between the qubits. We then sequentially pulse the qubits to drive one qubit conditionally on the state of the other. We find that a conditional $π$-r…
We consider the implementation of two-qubit unitary transformations by means of CNOT gates and single-qubit unitary gates. We show, by means of an explicit quantum circuit, that together with local gates three CNOT gates are necessary and sufficient in order to implement an arbitrary unitary transformation of two qubits. We also identify the subset of two-qubit gates that can be performed with only two CNOT gates.
We propose an implementation of the two-qubit gate in a quantum dot spin qubit system which is immune to charge noise problems. Our proposed implementation, if it could be realized in a physical system, would have the advantage of being robust against uncertainties and fluctuations in the tunnel coupling and barrier gate voltage pulse area. The key idea is to introduce an auxiliary dot and use an analog to the stimulated Raman adiabatic passage pulse sequence in three-level atomic systems, often referred to in the context of electron transport in quantum dot systems as Coherent Tunneling by Ad…
The number of two-qubit gates required to transform deterministically a three-qubit pure quantum state into another is discussed. We show that any state can be prepared from a product state using at most three CNOT gates, and that, starting from the GHZ state, only two suffice. As a consequence, any three-qubit state can be transformed into any other using at most four CNOT gates. Generalizations to other two-qubit gates are also discussed.
We analyze the cross-resonance effect for fluxonium circuits and investigate a two-qubit gate scheme based on selective darkening of a transition. In this approach, two microwave pulses at the frequency of the target qubit are applied simultaneously with a proper ratio between their amplitudes to achieve a controlled-NOT operation. We study in detail coherent gate dynamics and calculate gate error. With nonunitary effects accounted for, we demonstrate that gate error below $10^{-4}$ is possible for realistic hardware parameters. This number is facilitated by long coherence times of computation…
We demonstrate how using two-qubit composite rotations a high fidelity controlled-NOT (CNOT) gate can be constructed, even when the strength of the interaction between qubits is not accurately known. We focus on the exchange interaction oscillation in silicon based solid-state architectures with a Heisenberg Hamiltonian. This method easily applies to a general two-qubit Hamiltonian. We show how the robust CNOT gate can achieve a very high fidelity when a single application of the composite rotations is combined with a modest level of Hamiltonian characterisation. Operating the robust CNOT gate…
While mapping a quantum circuit to the physical layer one has to consider the numerous constraints imposed by the underlying hardware architecture. Connectivity of the physical qubits is one such constraint that restricts two-qubit operations, such as CNOT, to "connected" qubits. SWAP gates can be used to place the logical qubits on admissible physical qubits, but they entail a significant increase in CNOT-count. In this paper we consider the problem of reducing the CNOT-count in Clifford+T circuits on connectivity constrained architectures, like noisy intermediate-scale quantum (NISQ) computi…
Controlled-NOT (CNOT) gates are commonly included in the standard gate set of quantum processors and provide an important way to entangle qubits. For fixed-frequency qubits using the cross-resonance entangling technique, using the higher-frequency qubit to control the lower-frequency qubit enables much shorter entangling times than using the lower-frequency qubit as the control. Consequently, when implementing a CNOT gate where logical control by the lower-frequency qubit is needed, compilers may implement this functionality by using an equivalent circuit such as placing Hadamard gates on both…
Single qubit rotation gate and the controlled-NOT (CNOT) gate constitute a complete set of gates for universal quantum computation. In general the CNOT gate are only for two nearby qubits. For two qubits which are remote from each other, we need a series of swap gates to transfer these two qubits to the nearest neighboring sites, and then after the CNOT gate we should transfer them to their original sites again. However, a series of swap gates are resource for quantum information processing. One economy way which does not consume so much resource is to implement CNOT gate remotely. The remote …
We consider the CNOT quantum gate as a physical action, i.e. as unitary in time evolution of the two-qubit system. This points to the modeling of the interaction Hamiltonian of the two-qubit system which would correspond to the CNOT transformation; the analysis naturally generalizes to the Toffoli gate. Despite nonuniqueness of the model of the interaction Hamiltonian, the analysis distinguishes that the interaction Hamiltonian does not posses any global (rotational) symmetry. This forces us to conclude that the direct (non-mediated) interaction in the two-qubit system does not suffice for imp…
We give a simple, closed-form formula, what we call the Deflation Identity, for converting any 2-qubit circuit with exactly two controlled-U's (and some 1-qubit rotations) into an equivalent circuit with just two CNOTs (and some 1-qubit rotations). We also give two interesting applications of the Deflation Identity; one to "opening and closing a breach" in a quantum circuit, the other to the CS decomposition of a 2-qubit operator.
Implementing high-fidelity two-qubit gates in single-electron spin qubits in silicon double quantum dots is still a major challenge. In this work, we employ analytical methods to design control pulses that generate high-fidelity entangling gates for quantum computers based on this platform. Using realistic parameters and initially assuming a noise-free environment, we present simple control pulses that generate CNOT, CPHASE, and CZ gates with average fidelities greater than 99.99\% and gate times as short as 45 ns. Moreover, using the local invariants of the system's evolution operator, we sho…
Scalable and fault-tolerant quantum computation will require error correction. This will demand constant measurement of many-qubit observables, implemented using a vast number of CNOT gates. Indeed, practically all operations performed by a fault-tolerant device will be these CNOTs, or equivalent two-qubit controlled operations. It is therefore important to devise benchmarks for these gates that explicitly quantify their effectiveness at this task. Here we develop such benchmarks, and demonstrate their use by applying them to a range of differently implemented controlled gates and a particular…
The physical limitations of quantum hardware often require nearest-neighbor qubit structures, in which two-qubit gates are required to construct nearest-neighbor quantum circuits. However, two-qubit gates are considered a major cost of quantum circuits because of their high error rate as compared with single-qubit gates. The controlled-not (CNOT) gate is the typical choice of a two-qubit gate for universal quantum circuit implementation together with the set of single-qubit gates. In this study, we construct a one-dimensional nearest-neighbor circuit of quantum Fourier transform (QFT), which i…
In this note we present explicit canonical forms for all the elements in the two-qubit CNOT-Dihedral group, with minimal numbers of controlled-S (CS) and controlled-X (CX) gates, using the generating set of quantum gates [X, T, CX, CS]. We provide an algorithm to successively construct the n-qubit CNOT-Dihedral group, asserting an optimal number of controlled-X (CX) gates. These results are needed to estimate gate errors via non-Clifford randomized benchmarking and may have further applications to circuit optimization over fault-tolerant gate sets.
The three-input TOFFOLI gate is the workhorse of circuit synthesis for classical logic operations on quantum data, e.g., reversible arithmetic circuits. In physical implementations, however, TOFFOLI gates are decomposed into six CNOT gates and several one-qubit gates. Though this decomposition has been known for at least 10 years, we provide here the first demonstration of its CNOT-optimality. We study three-qubit circuits which contain less than six CNOT gates and implement a block-diagonal operator, then show that they implicitly describe the cosine-sine decomposition of a related operator. …
We derive a set of composite pulse sequences that generates CNOT gates and correct all systematic errors within the logical subspace to arbitrary order. These sequences are applicable for any two-qubit interaction Hamiltonian, and make no assumptions about the underlying noise mechanism except that it is constant on the timescale of the operation. We do assume access to error-free single-qubit gates, so single-qubit gate imperfections eventually limit the achievable fidelity. However, since single-qubit gates generally have much higher fidelities than two-qubit gates in practice, these pulse s…
We propose a scheme to construct a deterministic CNOT gate on static electron-spin qubits, allowing for deterministic scalable quantum computing in solid-state systems.The excess electron confined in a charged quantum dot inside a double-sided optical microcavity represents the qubit, and the single photons play a medium role. Moreover, our device can work in both the weak coupling and the strong coupling regimes, but high fidelities are achieved only when the ratio of the side leakage to the cavity loss is low. Finally, we assess the feasibility of this device and show it can be implemented w…
We give analytical solutions for the time-optimal synthesis of entangling gates between indirectly coupled qubits 1 and 3 in a linear spin chain of three qubits subject to an Ising Hamiltonian interaction with equal coupling $J$ plus a local magnetic field acting on the intermediate qubit. The energy available is fixed, but we relax the standard assumption of instantaneous unitary operations acting on single qubits. The time required for performing an entangling gate which is equivalent, modulo local unitary operations, to the $\mathrm{CNOT}(1, 3)$ between the indirectly coupled qubits 1 and 3…
There are many cases where the interaction between two qubits is not precisely known, but single qubit operations are available. In this paper we show how, regardless of an incomplete knowledge of the strength or form of the interaction between two qubits, it is often possible to construct a CNOT gate which has arbitrarily high fidelity. In particular, we show that oscillations in the strength of the exchange interaction in solid state Si and Ge structures are correctable.
A CNOT circuit is the key gadget for entangling qubits in quantum computing systems. However, the qubit connectivity of noisy intermediate-scale quantum (NISQ) devices is constrained by their {limited connectivity architecture}. To improve the performance of CNOT circuits on NISQ devices, we investigate the optimization of the size/depth of CNOT circuits under the limited connectivity architecture. We present a method that can optimize the size of any $n$-qubit CNOT circuit $O\left(\frac{n^2}{\log δ}\right)$ on any connected graph with minimum degree $δ$, and prove this bound is optimal for th…
We propose a deterministic and scalable scheme to construct a two-qubit controlled-NOT (CNOT) gate and realize entanglement swapping between photonic qubits using a quantum-dot (QD) spin in a double-sided optical microcavity. The scheme is based on spin selective photon reflection from the cavity and can be achieved in a nondestructive and heralded way. We assess the feasibility of the scheme and show that the scheme can work in both the weak coupling and the strong coupling regimes. The scheme opens promising perspectives for long-distance photonic quantum communication and distributed quantu…
The optimal cost of a three-qubit Fredkin gate is 5 two-qubit entangling gates, and the overhead climbs to 8 when restricted to controlled-not (CNOT) gates. By harnessing higher-dimensional Hilbert spaces, we reduce the cost of a three-qubit Fredkin gate from 8 CNOTs to 5 nearest-neighbor CNOTs. We also present construction of an n-control-qubit Fredkin gate with 2n+3 CNOTs and 2n single-qudit operations. Finally, we design deterministic and nondeterministic three-qubit Fredkin gates in photonic architectures. The cost of a nondeterministic three-qubit Fredkin gate is further reduced to 4 near…
Color-code quantum computation seamlessly combines Majorana-based hardware with topological error correction. Specifically, as Clifford gates are transversal in two-dimensional color codes, they enable the use of the Majoranas' nonabelian statistics for gate operations at the code level. Here, we discuss the implementation of color codes in arrays of Majorana nanowires that avoid branched networks such as T-junctions, thereby simplifying their realization. We show that, in such implementations, nonabelian statistics can be exploited without ever performing physical braiding operations. Physica…
A two-qubit controlled-NOT (CNOT) gate, realized by a controlled-phase (C-phase) gate combined with single-qubit gates, has been experimentally implemented recently for quantum-dot spin qubits in isotopically enriched silicon, a promising solid-state system for practical quantum computation. In the experiments, the single-qubit gates have been demonstrated with fault-tolerant control-fidelity, but the infidelity of the two-qubit C-phase gate is, primarily due to the electrical noise, still higher than the required error threshold for fault-tolerant quantum computation (FTQC). Here, by taking t…
It is shown that the two qubit CNOT (controlled NOT) gate can also be realised using q-deformed angular momentum states via the Jordan-Schwinger mechanism.Thus all the three gates necessary for universality i.e. Hadamard, Phase Shift and the two qubit CNOT gate are realisable with q-deformed oscillators.
We introduce a complete transformation rule set and a minimal equation set for controlled-NOT (CNOT)-based quantum circuits. Using these rules, quantum circuits that compute the same Boolean function are reduced to the same normal form. We can thus easily check the equivalence of circuits by comparing their normal forms. By applying the Knuth-Bendix completion algorithm to a set of modified 18 equations introduced by Iwama et al. 2002, we obtain a complete transformation rule set (i.e., a set of transformation rules with the properties of `termination' and `confluence'). Our transformation rul…
We prove that a generic three-qubit quantum logic gate can be implemented using at most 98 one-qubit rotations about the $y$- and $z$-axes and 40 CNOT gates, beating an earlier bound of 64 CNOT gates.
We propose a non-deterministic CNOT gate based on a quantum cloner, a quantum switch based on all optical routing of single photon by single photon, a quantum-dot spin in a double-sided optical microcavity with two photonic qubits, delay lines and other linear optical photonic devices. Our CNOT provides a fidelity of 78% with directly useful outputs for a quantum computing circuit and requires no ancillary qubits or electron spin measurements.
The anyonic excitations of topological two-body color code model are used to implement a set of gates. Because of two-body interactions, the model can be simulated in optical lattices. The excitations have nontrivial mutual statistics, and are coupled to nontrivial gauge fields. The underlying lattice structure provides various opportunities for encoding the states of a logical qubit in anyonic states. The interactions make the transition between different anyonic states, so being logical operation in the computational bases of the encoded qubit. Two-qubit gates can be performed in a topologic…
Entangling gates between qubits are a crucial component for performing algorithms in quantum computers. However, any quantum algorithm must ultimately operate on error-protected logical qubits encoded in high-dimensional systems. Typically, logical qubits are encoded in multiple two-level systems, but entangling gates operating on such qubits are highly complex and have not yet been demonstrated. Here, we realize a controlled NOT (CNOT) gate between two multiphoton qubits in two microwave cavities. In this approach, we encode a qubit in the high-dimensional space of a single cavity mode, rathe…
Efficient quantum circuits are necessary for realizing quantum algorithms on noisy intermediate-scale quantum devices. Fermionic excitations entering unitary coupled-cluster (UCC) ansätze give rise to quantum circuits containing CNOT "staircases" whose number scales exponentially with the excitation rank. Recently, Yordanov et al. [Phys. Rev. A 102, 062612 (2020); Commun. Phys. 4, 228 (2021)] constructed CNOT-efficient quantum circuits for both fermionic- (FEB) and qubit-excitation-based (QEB) singles and doubles and illustrated their usefulness in adaptive derivative-assembled pseudo-Trotteri…
In the framework of ion-trap quantum computing, we develop a characterization of experimentally realistic imperfections which may affect the Cirac-Zoller implementation of the CNOT gate. The CNOT operation is performed by applying a protocol of five laser pulses of appropriate frequency and polarization. The laser-pulse protocol exploits auxiliary levels, and its imperfect implementation leads to unitary as well as non-unitary errors affecting the CNOT operation. We provide a characterization of such imperfections, which are physically realistic and have never been considered before to the bes…
In this paper we present the complete simulation of the quantum logic CNOT gate in the one-way model, that consists entirely of one-qubit measurements on a particular class of entangled states.
We focus on the depth optimization of CNOT circuits on hardwares with limited connectivity. We adapt the algorithm from Kutin et al. that implements any $n$-qubit CNOT circuit in depth at most $5n$ on a Linear Nearest Neighbour (LNN) architecture. Our proposal is a block version of Kutin et al.'s algorithm that is scalable with the number of interactions available in the hardware: the more interactions we have the less the depth. We derive better theoretical upper bounds and we provide a simple implementation of the algorithm. Overall, we achieve better depth complexity for CNOT circuits on so…
Fewer-qubit quantum logic gate, serving as a basic unit for constructing universal multiqubit gates, has been widely applied in quantum computing and quantum information. However, traditional constructions for fewer-qubit gates often utilize a multi-pulse protocol which inevitably suffers from serious intrinsic errors during the gate execution. In this article, we report an optimal model about universal two- and three-qubit CNOT gates mediated by excitation to Rydberg states with easily-accessible van der Waals interactions. This gate depends on a global optimization to implement amplitude and…
We discuss a measurement-based implementation of a controlled-NOT (CNOT) quantum gate. Such a gate has recently been discussed for free electron qubits. Here we extend this scheme for qubits encoded in product states of two (or more) spins-1/2 or in equivalent systems. The key to such an extension is to find a feasible qubit-parity meter. We present a general scheme for reducing this qubit-parity meter to a local spin-parity measurement performed on two spins, one from each qubit. Two possible realizations of a multiparticle CNOT gate are further discussed: electron spins in double quantum dot…
We have embedded two fixed-frequency Al/AlO$_{\textrm{x}}$/Al transmons, with ground-to-excited transition frequencies at 6.0714 GHz and 6.7543 GHz, in a single 3D Al cavity with a fundamental mode at 7.7463 GHz. Strong coupling between the cavity and each transmon results in an effective qubit-qubit coupling strength of 26 MHz and a -1 MHz dispersive shift in each qubit's transition frequency, depending on the state of the other qubit. Using the all-microwave SWIPHT (Speeding up Waveforms by Inducing Phases to Harmful Transitions) technique, we demonstrate the operation of a generalized contr…
In this paper, we propose a new direction of research for the realization of the quantum controlled-not gate based on a technique called ``interaction-free measurement'', where qubits are two-level atoms (or ions) and information is mediated from one qubit to another by lasers in superpositions of $π$ and $2π$ pulses. We investigate the advantages and limitations of such a gate and discuss possible applicability.
Many physical implementations of quantum computers impose stringent memory constraints in which 2-qubit operations can only be performed between qubits which are nearest neighbours in a lattice or graph structure. Hence, before a computation can be run on such a device, it must be mapped onto the physical architecture. That is, logical qubits must be assigned physical locations in the quantum memory, and the circuit must be replaced by an equivalent one containing only operations between nearest neighbours. In this paper, we give a new technique for quantum circuit mapping (a.k.a. routing), ba…
Improving two-qubit gate performance and suppressing crosstalk are major, but often competing, challenges to achieving scalable quantum computation. In particular, increasing the coupling to realize faster gates has been intrinsically linked to enhanced crosstalk due to unwanted two-qubit terms in the Hamiltonian. Here, we demonstrate a novel coupling architecture for transmon qubits that circumvents the standard relationship between desired and undesired interaction rates. Using two fixed frequency coupling elements to tune the dressed level spacings, we demonstrate an intrinsic suppression o…
In the era of noisy intermediate-scale quantum (NISQ), executing quantum algorithms on actual quantum devices faces unique challenges. One such challenge is that quantum devices in this era have restricted connectivity: quantum gates are allowed to act only on specific pairs of physical qubits. For this reason, a quantum circuit needs to go through a compiling process called qubit routing before it can be executed on a quantum computer. In this study, we propose a CNOT synthesis method called the token reduction method to solve this problem. The token reduction method works for all quantum com…
