06/08/2021

NTT Basic Research Laboratories

Leonid V. Abdurakhimov, Imran Mahboob, Hiraku Toida, Kosuke Kakuyanagi, and Shiro Saito

Quantum computation is a new computing paradigm based on well-studied, but not yet fully exploited, quantum phenomena such as superposition and entanglement. The key element of a quantum computer is a qubit: a two-level system that can be prepared in an arbitrary state |*ψ* described by a linear combination (superposition) of qubit pure states and (i.e., , where *α* and *β* are complex amplitudes). The number of computational basis states of a system of *N* qubits scales with the number of qubits as 2^{N}. For example, a state of a two-qubit system can be described by a superposition of 2^{2} = 4 states (i.e., ). Some multi-qubit states can be represented as a product of individual qubit states: for example, the state of two qubits A and B can be decomposed to the product of the states of A and B × . Other multi-qubit quantum states—so-called entangled states—cannot be obtained by multiplication of single qubit states, for example, the two-qubit state . By using superposition and entanglement phenomena, information from many qubits can be simultaneously encoded in a single quantum state. Quantum algorithms, such as Shor’s algorithm for integer factorization and Grover’s search algorithm, use this property of quantum systems (quantum parallelism) to encode different function instances (different function realizations) in a single quantum state and compute their outcomes at once. A sufficiently large quantum computer will be able to solve a number of numerical problems, such as integer factorization and database search, much faster than conventional computers.

Several hardware platforms for quantum computing are being actively studied, including superconducting qubits, trapped ions, optical photonic systems, and spin qubits. Although the above hardware platforms have their own unique advantages and disadvantages, superconducting qubits is arguably the most advanced hardware platform for quantum computing, which is pursued by tech giants such as Google, IBM, and Microsoft, as well as new startups such as Rigetti Computing.

There are many types of superconducting qubits, but one particular variation (the so-called transmon) has become prevalent. Transmons can be roughly separated into two large groups: fixed-frequency and tunable qubits. Fixed-frequency qubits generally demonstrate better coherence times, while tunable qubits provide more freedom in terms of possible quantum protocols. In the most advanced superconducting qubit platforms, both fixed-frequency qubits (IBM, Yale University) and tunable qubits (Google, Delft University) are used (Table 1).

Despite the recent impressive progress in building small-scale superconducting qubit systems, development of a universal quantum computer is still a very challenging task. The major obstacle is the loss of quantum information due to the interaction with noisy environments. For single qubit operations, the important figures of merit are energy-relaxation time *T*_{1} and dephasing time *T*_{2}. In systems, in which quantum information is encoded directly into physical qubits (Google, IBM and Delft University systems), typical *T*_{1} and *T*_{2} values are in the range of 100 µs (Table 1), while characteristic qubit manipulation times are on the order of 100 ns. Therefore, error rates in state-of-the-art systems are on the order of *p* ≈ 10^{−3}. To execute reliable quantum computations at such error rates, implementation of error-correction algorithms is necessary, which will require about 10^{7} physical qubits. At the current level of technology, the brute-force approach to the scaling of superconducting qubit systems to such high qubit numbers is not straightforward. There is a limit to expanding the current qubit surface-mount technology, and next-generation mounting technologies such as distributed quantum computation are expected.

Recently, there has been significant interest in the idea of hardware-efficient quantum computing architectures, which use quantum systems with intrinsic protection against quantum errors. One of the promising approaches is bosonic quantum computing, which is based on the encoding of qubit states and into complex superposition states of a three-dimensional (3D) superconducting cavity. In the quantum regime, a 3D microwave cavity can be considered a harmonic oscillator with an infinite number of equidistant energy levels. Each level corresponds to a quantum state with a fixed number of microwave photons in the cavity (i.e., states correspond to one, two, …, *n* photons in the cavity). The pure cavity states are equidistant and cannot be used for qubit encoding because it is difficult to address only two of them without exciting other states. In bosonic computing, this issue is solved by encoding qubit states and into superposition cavity states, such as and , by using an additional “ancilla” superconducting qubit coupled to the cavity. The advantage of bosonic quantum computing is that the main error mechanism is the cavity-photon loss, which is relatively easy to detect and correct.

In this article, a new type of tunable qubit, 3D hybrid flux qubit, is described that can be used for bosonic computing. Due to their tunability, such qubits can be used as ancillas for achieving dynamical coupling between photonic states of different cavities, which will be an important step towards building a large-scale bosonic quantum computer.

Our 3D hybrid qubit consists of a capacitively shunted (c-shunt) 2D flux qubit embedded in a 3D copper cavity, as shown in Figs. 1 and 2 ^{[5]}. C-shunt flux qubits were proposed theoretically in 2007 ^{[6]} and were initially fabricated in a 2D architecture ^{[7]}. The novelty of our 3D hybrid qubit design relies on the combination of a 2D c-shunt flux qubit with a 3D microwave resonator.

A schematic and images of the 2D c-shunt flux qubit used as a part of our 3D hybrid qubit are shown in Figs. 1(a)–(d). The 2D qubit consists of an aluminum superconducting loop interrupted by three Josephson junctions (Figs. 1(a), (d)). Two junctions are identical, while the third junction is smaller than the other two. The small junction is shunted by a large coplanar capacitor consisting of two large rectangular aluminum pads (Figs. 1(a)–(c)). The 2D qubit is fabricated on a sapphire substrate, and Josephson junctions are fabricated using double-angle shadow evaporation of aluminum. Each Josephson junction consists of two superconducting aluminum layers separated by a thin (about 2 nm) insulating layer of aluminum oxide formed by oxidizing the aluminum film in oxygen atmosphere (Fig. 1(e)).

The substrate with the 2D c-shunt flux qubit is embedded in a 3D microwave cavity (Fig. 2(a)), and the device is cooled to 10 mK (approximately –273°C) using a dilution refrigerator. Because of the capacitive coupling between the qubit and cavity, the cavity-resonance frequency shifts to higher frequencies when the qubit is excited from the ground state to the excited state (Fig. 2(b)); hence, the state of the qubit can be determined from the cavity-resonance frequency. The qubit-transition frequency can be tuned by applying an external magnetic flux Φ_{e} = *B* × *S*, where *B* is the applied magnetic field and *S* is the qubit loop area (Fig. 2(c)).

This 3D hybrid flux qubit demonstrates long *T*_{1} in the range 60–90 µs ^{[5]} and *T*_{2} up to 100 µs ^{[8]} (Fig. 3(a)). These coherence times exceed those reported for other types of flux qubits (Fig. 3(b)) and are comparable to relaxation times observed in fixed-frequency qubits such as transmon qubits.

Our 3D hybrid flux qubit demonstrates relaxation times comparable to those reported for transmon qubits, which are typically used in superconducting platforms. This makes 3D hybrid flux qubits a possible alternative to transmons for applications in quantum computing. Due to its intrinsic 3D structure, this new design can be particularly useful for bosonic computing.

Quantum computing using bosonic states of 3D superconducting cavities is a promising approach for hardware-efficient quantum-information processing. The possibility of the error-correction of a single qubit encoded in cavity photon states has been recently demonstrated ^{[9]}. Next important steps will be improving the error-correction protocol to achieve full fault-tolerant protection and demonstrating logical operations (gates) between multiple error-corrected qubits. Two-qubit gates can be achieved by coupling qubits encoded in two cavities via ancilla superconducting qubits. For example, a controlled NOT (CNOT) gate between two cavity qubits using a fixed-frequency transmon qubit has been reported ^{[10]}. However, one of the problems of the transmon-based scheme is that the coupling between the cavities is fixed; therefore, there is an unwanted interaction between qubits encoded in cavity states even in the idle state. This issue can be solved using dynamical coupling schemes, in which the interaction between the cavities is switched on only for a short time. Our 3D hybrid flux qubit is uniquely suited for this purpose since its frequency can be adjusted by applying an external magnetic field. A possible concept of the dynamic coupling protocol using our 3D hybrid flux qubit is shown in Fig. 4. The coupling between cavity qubits is facilitated by two ancilla qubits capacitively coupled to each other (Fig. 4(a)). Ancillas are coupled to different cavities, and the interaction between the cavities can be switched on and off using the so-called iSwap gate ^{[11]} between ancilla qubits (Fig. 4(b)).

In conclusion, the new type of 3D superconducting qubit described in this article combines long coherence times with frequency tunability. Future work will explore possible applications of such high-coherence tunable qubits in hardware-efficient schemes of quantum computing, such as bosonic computing.

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