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arXiv:2607.11973

On the Experimental Determination of Nonlocal Characteristics of Two-Qubit Gates

M. Karthick Selvan, S. Balakrishnan

quant-phSubmitted Jul 13, 2026• Announced Jul 14, 2026• 0 citations
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Abstract

In this paper, we discuss the experimental determination of the nonlocal characteristics of two-qubit gates. Based on the recently derived expressions for the entangling power and gate typicality of two-qubit gates, we construct two-qubit quantum circuits to measure the entangling power and gate typicality of the two-qubit gates, which are generated by the elements of the su(4) Cartan subalgebra. These elements describe the native interactions in many quantum processors. Hence, these circuits can be used to determine the nonlocal characteristics of native gates of many quantum processors. In each circuit, the native gate is applied twice. In addition, each circuit consists of at least one CNOT gate. A set of six two-qubit circuits is constructed to measure the entangling power. The number of circuits is further reduced to three by increasing the nonlocal resources (the number of CNOT gates). To measure the gate typicality, a set of three two-qubit circuits is constructed. Measurement of gate typicality requires more nonlocal resources than the measurement of entangling power.

Comments
7 pages. Research in progress. Comments are welcome

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In this paper, we discuss the experimental determination of the nonlocal characteristics of two-qubit gates. Based on the recently derived expressions for the entangling power and gate typicality of two-qubit gates, we construct two-qubit quantum circuits to measure the entangling power and gate typicality of the two-qubit gates, which are generated by the elements of the su(4) Cartan subalgebra. These elements describe the native interactions in many quantum processors. Hence, these circuits can be used to determine the nonlocal characteristics of native gates of many quantum processors. In each circuit, the native gate is applied twice. In addition, each circuit consists of at least one CNOT gate. A set of six two-qubit circuits is constructed to measure the entangling power. The number of circuits is further reduced to three by increasing the nonlocal resources (the number of CNOT gates). To measure the gate typicality, a set of three two-qubit circuits is constructed. Measurement of gate typicality requires more nonlocal resources than the measurement of entangling power.
Abstract We show that the geometrical features of the unimodular squared eigenvalues of the nonlocal part of a two-qubit gate in the complex plane can be used to study the nonlocal characteristics of the two-qubit gate. It has been shown that the entangling power of a two-qubit gate can be expressed in terms of the mean squared length of the chords connecting the squared eigenvalues of the nonlocal part with each other. In this paper, we provide similar expressions for the gate typicality and the linear entropy of two-qubit gates. The expression of gate typicality involves the squared lengths of three chords connecting a squared eigenvalue of the nonlocal part with the complex conjugate of the other three squared eigenvalues. The expression of linear entropy is obtained from the expressions of entangling power and gate typicality. We also express them in terms of the linear entropies of certain two-qubit states. In addition, we show that the normalized entangling power of a two-qubit gate can be provided exact description in the Argand diagram. Finally, we analyze the simplices enclosing the origin in the convex hull of the squared eigenvalues of the nonlocal part of the perfect entangling two-qubit gates to construct a pair of orthonormal product states that can be transformed into maximally entangled states, and divide the region of perfect entanglers in the Weyl chamber into different regions based on the uniqueness of the simplices enclosing the origin.
Circuit cutting enables quantum computations to be decomposed into smaller subcircuits at the cost of additional sampling overhead. Gate-cutting provides one such approach by replacing nonlocal gates with quasiprobability decompositions of local operations. Starting from the known optimal two-qubit gate-cutting formula, we derive complete sharp lower and upper envelopes relating the quasiprobability extent γγγ to five established descriptors: entangling power, gate typicality, operator entanglement, Schmidt strength, and maximum product-input concurrence. We recast γγγ as a Rényi-1/21/21/2 functional of the operator-Schmidt spectrum and show how the physical constraints on two-qubit spectra sharpen generic entropy bounds and select a small set of recurring extremal Cartan families. None of the five descriptors generically determines γγγ alone, but each imposes exact constraints, revealing how substantially the optimal gate-cutting cost can vary between gates with similar nonlocal characteristics.
Abstract There are various gate sets that can be used to describe a quantum computation. A particularly popular gate set in the literature on quantum computing consists of arbitrary single-qubit gates and two-qubit CNOT gates. A CNOT gate is however not always the natural multi-qubit interaction that can be implemented on a given physical quantum computer, necessitating a compilation step that transforms these CNOT gates to the native gate set. An especially interesting case where compilation is necessary is for ion trap quantum computers, where the natural entangling operation can act on more than two qubits and can even act globally on all qubits at once. This calls for an entirely different approach to constructing efficient circuits. In this paper we study the problem of converting a given circuit that uses two-qubit gates to one that uses global gates. Our three main contributions are as follows. First, we find an efficient algorithm for transforming an arbitrary circuit consisting of Clifford gates and arbitrary phase gates into a circuit consisting of single-qubit gates and a number of global interactions proportional to the number of non-Clifford phases present in the original circuit. Second, we find a general strategy to transform a global gate that targets all qubits into one that targets only a subset of the qubits. This approach scales linearly with the number of qubits that are not targeted, in contrast to the exponential scaling reported in (Maslov and Nam 2018 New J. Phys. 20 033018). Third, we improve on the number of global gates required to synthesise an arbitrary n-qubit Clifford circuit from the 12n − 18 reported in (Maslov and Nam 2018 New J. Phys. 20 033018) to 6n − 8.
In circuit-based quantum computing the available gate set typically consists of single-qubit gates acting on each individual qubit and at least one entangling gate between pairs of qubits.In certain physical architectures, however, some qubits may be "hidden" and lacking direct addressability through dedicated control and readout lines, for instance, because of limited on-chip routing capabilities, or because the number of control lines becomes a limiting factor for many-qubit systems.In this case, no single-qubit operations can be applied to the hidden qubits and their state cannot be measured directly.Instead, they may be controlled and read out only via single-qubit operations on connected "control" qubits and a suitable set of two-qubit gates.We first discuss the impact of such restricted control capabilities on the performance of specific qubit coupling networks.We then experimentally demonstrate full control and measurement capabilities in a superconducting two-qubit device with local single-qubit control and iSWAP and controlledphase two-qubit interactions enabled by a tunable coupler.We further introduce an iterative tune-up process required to completely characterize the gate set used for quantum process tomography and evaluate the resulting gate fidelities.
The capacity of a quantum gate to produce entangled states on a bipartite system is quantified in terms of the entangling power. This quantity is defined as the average of the linear entropy of entanglement of the states produced after applying a quantum gate over the whole set of separable states. Here we focus on symmetric two-qubit quantum gates, acting on the symmetric two-qubit space, and calculate the entangling power in terms of the appropriate local invariant. A geometric description of the local equivalence classes of gates is given in terms of the su(3)\text{su}(3)su(3) Lie algebra root vectors. These vectors define a primitive cell with hexagonal symmetry on a plane, and through the Weyl group the minimum area on the plane containing the whole set of locally equivalent quantum gates is identified. We give conditions to determine when a given quantum gate produces maximally entangled states from separable ones (perfect entanglers). We find that these gates correspond to one-fourth of the whole set of locally distinct quantum gates. The formalism developed here is applicable to general three-level systems. Via the Majorana representation, qutrit transformations can be regarded as having entangling power and hence classified as perfect and nonperfect entanglers and be grouped into local-equivalence classes of the associated symmetric two-qubit space. The results are illustrated by an anisotropic Heisenberg model, the Lipkin-Meshkov-Glick model, and two coupled quantized oscillators with cross-Kerr interaction, which we use to obtain three-level gates relevant in qutrit quantum computation.
The speed of elementary quantum gates, particularly two-qubit gates, ultimately sets the limit on the speed at which quantum circuits can operate. In this work, we experimentally demonstrate commonly used two-qubit gates at nearly the fastest possible speed allowed by the physical interaction strength between two superconducting transmon qubits. We achieve this quantum speed limit by implementing experimental gates designed using a machine-learning-inspired optimal control method. Importantly, our method only requires the single-qubit drive strength to be moderately larger than the interaction strength to achieve an arbitrary two-qubit gate close to its analytical speed limit with high fidelity. Thus the method is applicable to a variety of platforms, including those with comparable single-qubit and two-qubit gate speeds, or those with always-on interactions. We expect our method to offer significant speedups for non-native two-qubit gates that are typically achieved with a long sequence of single-qubit and native two-qubit gates.