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C Superconducting Quantum Computer This appendix reviews the technology used to create the quantum data plane and the control and measurement plan for superconducting qubits. In this design, a superconducting resonator is coupled with a nonlinear inductor to form an artificial atom, and these âatomsâ are used as the qubits for the computer. C.1 FABRICATION Low loss requires superconductors: a unique class of materials that exhibit no electrical resistance at zero frequency (that is, for direct currents) when cooled to below a critical temperature, Tc. Qubits for digital quantum computing and quantum simulation are most commonly fabricated from aluminum wiring (Tc = 1.2 K) and aluminum-amorphous aluminum oxide-aluminum (Al-AlOx-Al) Josephson junctions on either silicon or sapphire substrates. While superconducting qubits can be fabricated using the same design tools and fabrication equipment used to build silicon chips, the premium placed on high coherence necessitates that the specific fabrication steps be modified to eliminate defects that create losses. As a result, the highest-coherence qubits fabricated todayâwith coherence times of around 100 microsecondsâare generally very simple devices, using a single layer of metal, rather than the complex processes of 10 metal layers used with the digital silicon or superconducting logic devices in todayâs classical computers. In contrast, commercial quantum annealing computers that feature in excess of 2,000 superconducting qubits are fabricated using a more complex technology. This technology uses niobium wiring (Tc = 9.2 K) and niobium-amorphous aluminum oxide-niobium (Nb/AlOx/Nb) Josephson junctions [1,2] in a process that supports up to eight metal layers. This more complex fabrication process enables the qubits and superconducting control electronics to be integrated together in a single niobium fabrication process (an instance of âmonolithic integrationâ). However, due to the fabrication complexity, additional processing steps, and the need for an interwiring layer of dielectric materials like silicon dioxide or silicon nitride that cause loss, qubits made in multilayer niobium processes generally have low coherence times, typically in the 10-100 nanosecond range . C.2 QUBIT DESIGN Like a trapped ion qubit, a superconducting qubit can exist in a series of quantized energy states; the two lowest states can be accessed selectively to realize the qubit. Rather than using an atom, this design uses a simple inductor and capacitor circuit, which also has quantized energy at low temperatures. To make the energy difference between its levels distinct, a nonlinear inductive element, the Josephson junction (JJ) is added to the circuit. With a JJ, the difference between the ground state and the first excited state may be uniquely addressed by a frequency f01. This means that the microwave radiation, typically designed to be around 5 GHz, can be used to cause transitions between these two states without accessing the higher-excited states. Thus, this structure can be used as a qubit: a two-level quantum system. There are a number of ways the inductor, capacitor, and JJ can be arranged to create a qubit, and how the qubits are connected to each other to enable two-qubit operations. These differences trade off between simpler control and better isolation and control of qubit operations, as follows: PREPUBLICATION COPY â SUBJECT TO FURTHER EDITORIAL CORRECTION C-1
ï· Fixed-frequency versus tunable qubits. Frequency-tunable qubits can be calibrated and corrected for qubit frequency variations that arise from variations in the fabrication process or as a result of device aging. An advantage is that one microwave tone can control multiple qubits, a savings in hardware. Gaining this advantage requires an additional control signal to adjust the frequency and adds an additional path for noise to enter the qubit. The two most common qubits in use today for digital superconducting quantum computing are the âtransmon qubit,â[4,5,6,7] which comes in single-junction nontunable and two-junction tunable forms, and the âflux qubitâ [8-11]. Both transmon designs are being used in leading edge efforts. ï· Static versus tunable coupling. Static coupling between qubitsâfor example, by using a capacitor or an inductor to mediate interactionâis an âalways-onâ coupling that is fixed by design. The coupling is turned âonâ by bringing two qubits into resonance, and it is turned off by detuning the qubits. Yet even in the off state, there still is a small residual coupling. This tuning can be further reduced by adding a third objectâeither another coupler qubit or a resonatorâbetween the two qubits. The two qubits are then coupled by adjusting the qubits and the resonator to the proper frequency. In addition to the qubits, the circuits include a simple mechanism to couple the qubit to its 5 GHz microwave control signal and to a superconducting resonator, typically designed to operate at around 7-8 GHz, which reads out the qubit state using the circuit quantum electrodynamics architecture . C.3 REFRIGERATION Superconducting qubits require milli-Kelvin (mK) temperatures to operate. For digital quantum computing, the qubit operation frequency is typically around 5 GHz, which corresponds to a thermal energy of approximately 250 mK; the qubit must thus be operated at much lower temperatures in order to avoid unwanted thermal excitation of the excited state. This is achieved using commercial 3He/4He dilution refrigerators, which are capable of cooling to sub-10 mK temperatures. On the other hand, for most practical potential uses of a quantum annealer, the qubits will at times operate at frequencies corresponding to thermal temperatures much lower than those achievable with a dilution refrigerator, which make it nearly certain that thermal noise will affect the annealing protocol and drive the system out of its ground state. Modern dilution refrigerators leverage electromechanical pulse-tube coolers to achieve cooling in two stages, one at 50 K and one at 3 K. These are called âdryâ refrigerators, as they do not require consumable liquid helium coolant to reach these temperatures. Then, at 3 K, a closed-cycle mixture of helium isotopesâ3He and 4Heâis condensed and circulated to achieve cooling through a series of stages at temperatures of 700 mK, 50 mK, and the base temperature of approximately 10 mK. Cooling from room temperature to base temperature generally takes about 36 to 48 hours, and the refrigerator can remain cold indefinitely. In contemporary commercial dilution refrigerators, the experimental volume at base temperature is about (0.5 m)3 and the cooling power at base temperature / 20 mK / 100 mK is approximately 0 (by definition) / 30 Î¼W / 1000 W, respectively. These are not fundamental limits. Large objects in excess of 1 ton have been cooled to less than 10 mK using a dry dilution refrigerator for the CUORE neutrino detection experiment . Each temperature stage comprises a copper plate of approximately 0.5 m diameter, and they are used to thermalize control wiring from room temperature to base temperature both to cool the wires and to reduce thermal radiation from reaching the qubits . Coaxial cables, attenuators, filters, isolators/circulators, and microwave switches work at cryogenic temperatures and are all used in state-of-art measurement systems. PREPUBLICATION COPY â SUBJECT TO FURTHER EDITORIAL CORRECTION C-2
C.4 CONTROL AND MEASUREMENT PLANE The control and measurement plane for a superconducting quantum computer needs to generate the bias voltages/currents used to tune the qubits, create the microwave control signals, and reliably detect qubit measurements, while dealing with the large temperature differences that exist between the circuits that generate the control signals and the quantum plane that consumes them. C.4.1 Control Wiring and Packaging The delivery of electromagnetic control signals from the room-temperature region where they are generated to the qubits inside the refrigerator at mK temperatures requires careful thermal and electrical engineering. Wiringâwhether low-frequency twisted pairs or high-frequency coaxâmust be thermalized at each temperature stage of the refrigerator to avoid excessive heating of the mixing chamber. Perhaps counterintuitively, the thermal heating of the refrigerator through direct contact (phonons) is not the critical challenge. The largest heat loads occur across the 300 to 3 K transition, and todayâs refrigerators can readily handle the heat loads of hundreds and even thousands of wires. And, as larger wire counts are needed, larger dilution refrigerators with additional cooling at all stagesâin particular, at the 3 K stageâ can be built as a straightforward extension of existing technology at proportional cost. For the 3 K to milli-K wires, superconducting NbTi can deliver the electrical signals faithfully, with minimal heating due to the direct thermal connection (phonons). A more important challenge is mitigating the effects of room-temperature thermal noise on the operation of the qubits. There is a trade-off between efficiently guiding a desired signal to a qubit and preventing noise from impacting its operation. A two-pronged approach is used. Filtering (attenuating signals that are not in the range of desired frequencies) is used to remove out-of-band radiationânoise that is outside the frequency range of the signals intended to be delivered to the deviceâbut attenuation must be used to reduce the in-band radiation. This means that the amplitude of the control signal is decreased at each stage in the refrigerator, since the size of the thermal noise decreases with temperature. The attenuating cannot all be done at one point, since signal attenuation generates heat and thermal noise that must also decrease as the signal moves to lower temperatures. For similar reasons, the measurement of the qubit must also be done in stages, with the first stages of amplification performed at cryogenic temperatures, to minimize the noise of the amplifier. One critical constraint in chips with a large number of signals is packaging. The package for a supercomputing chip must house, shield, and route signals to/from a qubit chip; it is a critical part of the control plane. While the superconducting chips are relatively smallâtypically 5 Ã 5 mm2âit is the number of wires that feed the chip and their connectors that dictate the size of the package. For the high isolation needed for quantum circuits, coaxial connectors, coaxial wiring harnesses, miniature multipin connectors, and so on are types of connectors being used to bring signals into the package. The higher isolation that these connectors provide make them larger than the simple pin or ball connection used in packages for conventional silicon devices, and thus the number of signals per unit area is much smaller. Once the signals are on the package, they need to be routed to the correct location and then connected to the quantum circuit. Signals are connected to the qubit via wires, using bump (connections over the area of the chip) or wire (connections around the perimeter of the chip) bonds , or through the free-space of the package itself . As the number of control wires increase, these packages will need to move to area bonding methods (bump bonding) like what was done with conventional silicon packaging. The challenge is to maintain a clean microwave environment for the qubits in the presence of these connectors and wiring. Given these constraints, the packaging problem will become very difficult as the number of signals increase to the thousands. PREPUBLICATION COPY â SUBJECT TO FURTHER EDITORIAL CORRECTION C-3
C.4.2 Control and Measurement Having established a means to transfer signals between room temperature and the quantum data plane, the control and measurement layer needs to provide the hardware and software to (1) bias the qubit at its operating point; (2) perform logic operations; and (3) measure the qubit state. Contemporary superconducting qubits are operated using a combination of DC bias currents, microwave pulses resonant with the qubit transitionâtypically around 5 GHzâand baseband pulses. As was mentioned earlier, qubits can be either âfixed frequencyâ or âtunable frequency.â In a fixed-frequency design, the fabrication sets the qubit frequency, and the measurement system must determine that frequency and adjust its signals to it. The base frequency of tunable qubits is also set during fabrication, but it can be adjusted in situ using a bias current from the control plane. This bias current is connected through the qubit package and then coupled into the desired qubit. Tunable qubits require an extra control line but allow the control system to use a single frequencyâor a small set of frequenciesâfor all qubits. Control signals for single-qubit and two-qubit logic operations are generated using a stable microwave source, a programmable pulse shape, and a mixer, which combines the two signals to produce the needed microwave pulse. These pulses are around 10 ns (10 billionths of a second), generally much faster than those used for trapped ion qubits. Combinations of microwave pulses and frequency offsets are used to achieve two-qubit gate operationâfor example, a controlled-phase gate or an iSWAP gate. These gates are slower than single-qubit operations and take between 40 ns and 400 ns. The exact control signals depend on whether the qubits are directly coupled or use an additional qubit or resonator to minimize background coupling. State-of-art two-qubit error rate is generally at the 1 percent level, with individual examples as low as 0.5 percent. The requisite room-temperature control electronicsâmicrowave oscillators, arbitrary waveform generators (AWGs) to generate the pulse shapes, mixers, and analog-to-digital converters (ADCs)âare all commercially available items with sufficient precision to not limit the qubit operation. For contemporary superconducting qubit applications, the AWGs and ADCs typically operate with 1-2 GS/s and 10-14 bits of resolution. Commercially available precision-grade local oscillators typically have a 1- 12 GHz frequency range with a single-sideband phase noise of â120 dB at 10 kHz offset; this level of phase is generally sufficient to achieve gate error rates at the 10-8 level . As the number of qubits increases, the support electronics grow as well. Generally, there are bias current generators, waveform generators, and mixers needed for each qubit. Thus, there is a need to better integrate this support electronics to enable the systems to scale to larger number of qubits. Unlike natural atoms, which are all identical, artificial atoms are built from circuit elements, which have manufacturing variations. Thus, the qubit parameters (e.g., the transition frequency, qubit- qubit coupling, etc.) will differ from qubit to qubit, from one manufactured device to another, and from one temperature cycle to another. The control processor must have extensive calibration routines, to first determine, and then compensate for these variations. The complexity of this calibration grows superlinearly with the number of qubits in the system, and is one of the critical issues in scaling up the number of qubits. PREPUBLICATION COPY â SUBJECT TO FURTHER EDITORIAL CORRECTION C-4
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