Although only two quantum states of a physical system are often used to encode quantum information in the form of qubits, many levels can in principle be used to obtain qudits and increase the information capacity of the system. To take advantage of the additional levels, a parameterization of unitary transformations in terms of experimentally realizable operations is needed. Many parameterizations of unitary 3 * 3 matrices (U(3)) exist. One decomposition of a general unitary matrix can be expressed as the product of an exponential of a diagonal matrix and an exponential of an off-diagonal matrix. This decomposition is relevant for controlling superconducting qutrits using fixed-frequency resonant control pulses. This decomposition is numerically confirmed to allow the parameterization of any element in U(3). It is shown that a simple setting of parameter ranges of parameters can easily lead to an over-parameterization, in the sense that several different sets of values for the parameters produce the same element in U(3). This fact is demonstrated using the Walsh-Hadamard (WH) matrix as an example, which is also a special qutrit gate of practical interest. The different decompositions are shown to be related, and the relationships between them are presented using general methods. The shortest path needed for the implementation of a qutrit gate is found. Other parameterizations obtained by other analytic means, which can be advantageous for various reasons, are also discussed.
Although only two quantum states of a physical system are often used to encode quantum information in the form of qubits, many levels can in principle be used to obtain qudits and increase the information capacity of the system. To take advantage of the additional levels, a parameterization of unitary transformations in terms of experimentally realizable operations is needed. Many parameterizations of unitary 3 * 3 matrices (U(3)) exist. One decomposition of a general unitary matrix can be expressed as the product of an exponential of a diagonal matrix and an exponential of an off-diagonal matrix. This decomposition is relevant for controlling superconducting qutrits using fixed-frequency resonant control pulses. This decomposition is numerically confirmed to allow the parameterization of any element in U(3). It is shown that a simple setting of parameter ranges of parameters can easily lead to an over-parameterization, in the sense that several different sets of values for the parameters produce the same element in U(3). This fact is demonstrated using the Walsh-Hadamard (WH) matrix as an example, which is also a special qutrit gate of practical interest. The different decompositions are shown to be related, and the relationships between them are presented using general methods. The shortest path needed for the implementation of a qutrit gate is found. Other parameterizations obtained by other analytic means, which can be advantageous for various reasons, are also discussed.
We have implemented a Walsh-Hadamard gate, which performs a quantum Fourier transform, in a superconducting qutrit. The qutrit is encoded in the lowest three energy levels of a capacitively shunted flux device, operated at the optimal flux-symmetry point. We use an efficient decomposition of the Walsh-Hadamard gate into two unitaries, generated by off-diagonal and diagonal Hamiltonians, respectively. The gate implementation utilizes simultaneous driving of all three transitions between the three pairs of energy levels of the qutrit, one of which is implemented with a two-photon process. The gate has a duration of 35 ns and an average fidelity over a representative set of states, including preparation and tomography errors, of 99.2%, characterized with quantum-state tomography. Compensation of ac-Stark and Bloch-Siegert shifts is essential for reaching high gate fidelities.
A general scheme is presented to decompose a d-by-d unitary matrix as the product of two-level unitary matrices with additional structure and prescribed determinants. In particular, the decomposition can be done by using two-level matrices in d - 1 classes, where each class is isomorphic to the group of 2 × 2 unitary matrices. The proposed scheme is easy to apply, and useful in treating problems with the additional structural restrictions. A Matlab program is written to implement the scheme, and the result is used to deduce the fact that every quantum gate acting on n-qubit registers can be expressed as no more than 2n-1(2n-1) fully controlled single-qubit gates chosen from 2n-1 classes, where the quantum gates in each class share the same n - 1 control qubits. Moreover, it is shown that one can easily adjust the proposed decomposition scheme to take advantage of additional structure evolving in the process.
Physical quantum systems are commonly composed of more than two levels and offer the capacity to encode information in higher-dimensional spaces beyond the qubit, starting with the three-level qutrit. Here, we encode neutral-atom qutrits in an ensemble of ultracold 87Rb and demonstrate arbitrary single-qutrit SU(3) gates. We generate a full set of gates using only two resonant microwave tones, including synthesizing a gate that effects a direct coupling between the two disconnected levels in the three-level $\mathrm{\ensuremath{\Lambda}}$-scheme. Using two different gate sets, we implement and characterize the Walsh-Hadamard Fourier transform, and find similar final-state fidelity and purity from both approaches. This work establishes the ultracold neutral-atom qutrit as a promising platform for qutrit-based quantum information processing, extensions to d-dimensional qudits, and explorations in multilevel quantum state manipulations with nontrivial geometric phases.
Decomposing unitary operations into native gates is an essential step for implementing quantum algorithms. For qubit-based devices, where native gates are typically single- and two-qubit operations, a range of decomposition techniques have been developed. In particular, efficient algorithms exist for decomposing exponentials of Pauli strings while taking hardware topology in account. Motivated by the growing interest in qutrit-based quantum computing, we develop analogous decomposition methods for qutrit systems. Specifically, we introduce an algorithm that decomposes the exponential of an arbitrary tensor product of Weyl-Heisenberg operators (plus their Hermitian conjugation) into single- and two-qutrit gates. We further extend this approach to unitaries generated by Gell-Mann string (i.e., a tensor product of Gell-Mann matrices). Since both Gell-Mann matrices and Weyl-Heisenberg operators form (together with identity) complete operator bases of qutrit operators, we can use this result also to decompose any multi-qutrit gate that is diagonal up to single-qutrit rotations. As a practical application, we use our method to decompose the layers of the quantum approximate optimization algorithm for qutrit-based implementations of the graph k-coloring problem. For values of k well-suited to qutrit architectures (e.g., k=3 or in general k=3n), our approach yields significantly shallower circuits compared to qubit-based implementations, an advantage that grows with problem size, while also requiring a smaller total Hilbert space dimension. Finally, we also address the routing challenge in qutrit architectures that arises due to the limited connectivity of the devices. In particular, we generalize the Steiner-Gauss method, originally developed to reduce CNOT counts in qubit circuit, to optimize gate routing in qutrit-based systems.
The use of a few intermediate qutrits for efficient decomposition of 3-qubit unitary gates has been proposed, to obtain an exponential reduction in the depth of the decomposed circuit. An intermediate qutrit implies that a qubit is operated as a qutrit in a particular execution cycle. This method, primarily for the NISQ era, treats a qubit as a qutrit only for the duration when it requires access to the state ∣2⟩ during the computation. In this article, we study the challenges of including fault-tolerance in such a decomposition. We first show that any qubit that requires access to the state ∣2⟩ at any point in the circuit, must be encoded using a qutrit quantum error correcting code (QECC), thus resulting in a circuit with both qubits and qutrits at the outset. Since qutrits are noisier than qubits, the former is expected to require higher levels of concatenation to achieve a particular accuracy than that for qubit-only decomposition. Next, we derive analytically (i) the number of levels of concatenation required for qubit-qutrit and qubit-only decompositions as a function of the probability of error, and (ii) the criterion for which qubit-qutrit decomposition leads to a lower gate count than qubit-only decomposition. We present numerical results for these two types of decomposition and obtain the situation where qubit-qutrit decomposition excels for the example circuit of the quantum adder by considering different values for quantum hardware-noise and non-transversal implementation of the 2-controlled ternary CNOT gate.
We introduce a scheme for realizing arbitrary controlled-unitary operations in a two-qubit system. If the $2\ifmmode\times\else\texttimes\fi{}2$ unitary matrix is special unitary (has unit determinant), the controlled-unitary gate operation can be realized in a single pulse operation. The pulse in our scheme will constitute varying one of the parameters of the system between an arbitrarily maximum and a ``calculated'' minimum value. This parameter will constitute the variable parameter of the system while the other parameters, which include the coupling between the two qubits, will be treated as fixed parameters. The values of the parameters are what we solve for using our approach in order to realize an arbitrary controlled-unitary operation. We further show that the computational complexity of the operation is no greater than that required for a controlled-NOT (CNOT) gate. Since conventional schemes realize a controlled-unitary operation by breaking it into a sequence of single-qubit and CNOT gate operations, our method is an improvement because we not only require less time duration, but also fewer control lines to implement the same operation. To demonstrate improvement over other schemes, we show, as examples, how two controlled-unitary operations, one being the controlled-Hadamard gate, can be realized in a single pulse operation using our scheme. Furthermore, our method can be applied to a wide range of coupling schemes and can be used to realize gate operations between two qubits coupled via Ising, Heisenberg, and anisotropic interactions.