- Research
- Open Access
Engineering cryogenic setups for 100-qubit scale superconducting circuit systems
- S. Krinner^{1}Email authorView ORCID ID profile,
- S. Storz^{1},
- P. Kurpiers^{1},
- P. Magnard^{1},
- J. Heinsoo^{1},
- R. Keller^{1},
- J. Lütolf^{1},
- C. Eichler^{1} and
- A. Wallraff^{1}
- Received: 21 November 2018
- Accepted: 4 April 2019
- Published: 28 May 2019
Abstract
A robust cryogenic infrastructure in form of a wired, thermally optimized dilution refrigerator is essential for solid-state based quantum processors. Here, we engineer a cryogenic setup, which minimizes passive and active heat loads, while guaranteeing rapid qubit control and readout. We review design criteria for qubit drive lines, flux lines, and output lines used in typical experiments with superconducting circuits and describe each type of line in detail. The passive heat load of stainless steel and NbTi coaxial cables and the active load due to signal dissipation are measured, validating our robust and extensible concept for thermal anchoring of attenuators, cables, and other microwave components. Our results are important for managing the heat budget of future large-scale quantum computers based on superconducting circuits.
1 Introduction
Promising solid-state based quantum computing platforms, such as superconducting circuits [1] or charges and spins in semiconductor quantum dots [2], require temperatures on the Millikelvin level to initialise the systems in their ground state and to avoid errors due to thermal excitation during operation. Millikelvin temperatures are achieved in \(\mathrm{He}^{3}/\mathrm{He}^{4}\) dilution refrigerators (DR). When scaling those approaches from the few qubit level to large scale quantum processors, an increasing number of microwave and DC cables need to be integrated into the dilution refrigerator. They connect the classical control electronics at room temperature (RT) to the quantum processor at the lowest temperature stage of the dilution refrigerator, creating a substantial heat load on the dilution refrigerator due to heat conduction. Besides this passive load, active load due to the dissipation of control signals in cables and attenuators plays a major role. Engineered dissipation is necessary to thermalize the incoming radiation fields and to reduce the number of thermal photons incident on the sample [3, 4].
While superconducting quantum processors operating 20 qubits have been realized [5, 6], their cryogenic setup and extensibility towards larger system sizes has not been reported on. Here, we present a thermally optimized, robust cabling scheme and cryogenic setup suitable for the operation of 50 qubits at a temperature of 14 mK. Disregarding space constraints in the dilution refrigerator, at least 150 qubits could be operated using our approach. Before describing our customized cryogenic system in Sect. 3, we put forward methods for minimizing passive heat load on all stages of the dilution refrigerator and for minimizing the number of thermal photons in cables connecting room temperature components of the setup to the base temperature stage while keeping the active load low (Sect. 2). We then present measurements of the passive load arising from the installed cable trees comprising typically 25 coaxial lines each, and compare them to estimates based on available data on thermal conductances of coaxial cables (Sect. 4). Active load due to application of control signals is characterized in Sect. 5. Finally, we discuss the total heat load in Sect. 6 along with possible improvements which we expect to allow to increase the number of operational qubits to up to one thousand when disregarding space constraints.
1.1 Typical cabling for experiments with superconducting circuits
When designing the control lines and output lines connecting to a superconducting quantum processor the goal is to provide sufficiently strong coupling rates to the quantum processor, while minimizing decoherence due to coupling of the quantum processor via these lines to its environment. Thermal noise, present due to the connection of the quantum processor to electronics at room temperature, not only leads to qubit dephasing [3, 4, 7], but can also lead to creation of quasi-particles and thus to dissipation and reduced energy relaxation times [8–13]. Hence, thorough thermalization of cables, attenuators, and microwave components at the various temperature stages of the dilution refrigerator is not only important for reducing the heat load on the dilution refrigerator, but also for protecting the quantum processor from thermal radiation. In addition to thermal anchoring, filters with stop-bands outside the frequency range of qubits and readout resonators as well as infra-red blocking filters [12, 14, 15] further suppress thermal radiation. Other sources of external noise that can negatively impact coherence times include 1/f noise [7, 16] from electronic instrumentation and magnetic field fluctuations or magnetic offset fields inducing magnetic vortices that can lead to dissipation [12, 17, 18]. These sources can be mitigated with appropriate filters and magnetic shielding respectively, but their study is not at the focus of the present work.
We briefly present an overview of the cabling typically used in experiments with superconducting circuits. We distinguish between direct-current (DC) and radio-frequency (RF) cabling. DC lines are made from twisted pairs of wires, that are low-pass filtered, and thermalized at each temperature stage. Typical uses are biasing of cryogenic amplifiers and flux biasing of frequency-tunable qubits [19]. RF lines on the other hand are realized as semi-rigid microwave cables and contain various microwave components such as attenuators, filters and amplifiers. They connect to the quantum processor and are used for its control and readout. One typically distinguishes between drive lines, flux lines, and output lines, which we briefly describe here.
Drive lines are used for controlling the quantum states of qubits with a microwave tone realizing single-qubit gates, and for probing the frequency shift of readout resonators. To reduce thermal population of qubits, and frequency shifts of the qubits due to their dispersive interaction with a readout resonator [20], the number of thermal noise photons in the drive lines arriving at the mixing chamber plate (MXC) of the dilution refrigerator is required to be well below the single photon level in both cases. More precisely, to guarantee a noise photon number at MXC on the 10^{−3} level, a total attenuation of about 60 dB is required, see Sect. 2.2.1 and 3.2. At the same time, the bandwidth of the drive lines is required to be large enough to cover the typical frequency ranges of qubits (4–6 GHz) and of readout resonators (4–8 GHz).
Flux lines are used for implementing two-qubit gates which are based on the dynamical flux tunability of the transition frequency of a qubit [21–23] or of a separate coupling subcircuit [24, 25]. In addition, qubit frequency variations, occurring due to imperfections in the fabrication of Josephson junctions, can be compensated. Tunable frequency qubits, as opposed to fixed frequency qubits, make use of a SQUID (superconducting quantum interference device) loop instead of a single Josephson junction as the inductive element. By threading a magnetic flux Φ through this loop the frequency of the qubit approximately changes as \(\omega_{\mathrm{q}}\simeq\omega_{0} \sqrt{|\operatorname{cos}( \pi\varPhi/\varPhi_{0})|}\) [16], where \(\varPhi_{0}\) is the magnetic flux quantum. A mutual inductance between flux line and SQUID loop is realized by routing the on-chip part of the flux line past the SQUID loop. Hence, a current applied to the flux line results in a magnetic flux in the SQUID loop, effectively tuning the transition frequency of the qubit. Low-pass filters in the flux lines limit the bandwidth to about 1 GHz eliminating thermal noise at qubit frequencies. However, since the magnetic flux Φ in the SQUID loop sets the qubit transition frequency, any current noise leads to qubit dephasing [7, 16]. To reduce the current noise, a suitable amount of attenuation (10–20 dB) is also added in the flux lines, see Sect. 3.3.
Output lines contain a series of cryogenic and room temperature amplifiers for the detection of readout signals [26–28]. To isolate the sample from thermal noise photons and from input noise of amplifiers while not attenuating the output signal, isolators and circulators are used, see Sect. 3.4 for more details.
2 Sources of heat loads
We consider three dominant contributions to the heat load on the dilution refrigerator. First, passive load is due to heat flow from higher temperature stages to lower temperature stages. Here, we consider only heat conducted through installed cables. Heat which flows via posts that separate the various plates of the dilution refrigerator is not considered because it is already taken into account in the available cooling power of the dilution refrigerator specified by the manufacturer and separately evaluated in this work [29–31]. Second, active load arises due to the dissipation (Joule heating) of applied microwave signals in attenuators and in the microwave cables themselves. Dissipation arising from DC signals, used e.g. to bias HEMT amplifiers at the 4 K stage or to flux bias qubits on the chip at MXC, falls also in this category. Third, a radiative load arises due to blackbody radiation from stages and shields of higher temperature impinging on stages and shields of lower temperature [32]. This load, however, is also taken into account in the specified available cooling power of the dilution refrigerator. For completeness, we mention load due to residual gas, in particular Helium since it has the highest vapor pressure at cryogenic temperatures. However, during normal operation of the dilution refrigerator, the cryo-pumping capacity of the cold surfaces in the dilution refrigerator keeps the pressure in the vacuum can below 10^{−5} mbar providing an adequate isolation vacuum [32]. We therefore do not consider this load. However, latent heat released during desublimation processes can create additional loads, e.g. when gas desorbs from installed components and freezes out at the closest stage.
2.1 Passive load
To minimize passive heat load we use cable materials with low thermal conductivity. We note that with the exception of superconductors this typically goes along with poor electrical conductivity, and hence leads to more dissipation when microwave signals are applied. However, in most of the lines attenuation and thus dissipation is desired anyway to thermalize the incoming radiation fields, see Sect. 2.2.
Dilution refrigerator specifications. Temperatures and available cooling powers on the indicated stages of a Bluefors XLD400 DR. Coaxial cable lengths towards the respective stages are listed as well
Stage name | Temperature (K) | Cooling power (W) | Cable length (mm) |
---|---|---|---|
50 K | 35 | 30 (at 45 K) | 200 |
4 K | 2.85 | 1.5 (at 4.2 K) | 290 |
Still | 882 × 10^{−3} | 40 × 10^{−3} (at 1.2 K) | 250 |
CP | 82 × 10^{−3} | 200 × 10^{−6} (at 140 mK) | 170 |
MXC | 6 × 10^{−3} | 19 × 10^{−6} (at 20 mK) | 140 |
Three typical and readily available types of 0.085″ diameter coaxial cable are discussed. The lengths of the cables are listed in Table 1 and correspond to the inter-plate distances in the dilution refrigerator plus a small correction due to the bends used for stress and strain relief in those cables (see Sect. 3). Stainless steel cables (UT-085-SS-SS) and niobium-titanium cables (UT-085-NbTi) have the lowest passive load. Therefore, we choose stainless steel cables for the drive lines where large attenuation is needed. NbTi cable is superconducting below 10 K and therefore has very low attenuation below 10 K [36], which is why we use it for the output lines in the sections between the 4 K and MXC stages. The heat flows associated with these two cable types are dominated by their outer conductor, which has a cross-sectional area that is by a factor 10 larger than the cross-sectional area of the center conductor. The contribution of the Teflon dielectric is of the same order of magnitude as the one of the inner conductor, which is due to the low thermal conductivity of Teflon, see Appendix 1, and the cross-sectional area of the dielectric being comparable to the one of the outer conductor.
An alternative to stainless steel is cupronickel (CuNi) cable. It is expected to have a passive load which is about 50% larger than the one of SS-SS cable [37]. Another commonly used cable type is UT-085-SS cable (stainless steel outer conductor, SPCW^{1} center conductor), which results in a significantly larger heat load due to the larger thermal conductivity of Cu, which is anomalously enhanced at temperatures below 50 K, see Appendix 1. Due to its lower attenuation, see Appendix 2, this cable type is in principle suited for output lines in the sections from the vacuum flange of the dilution refrigerator to the 4 K stage, see discussion in Sect. 3.4.
In our dilution refrigerator the DC wiring was pre-installed by the manufacturer. Copper or phosphor-bronze (PhBr) twisted pairs of diameters AWG35 and AWG36 respectively are only used from room temperature to the 4 K stage due to their large thermal conductivity. From 4 K to MXC superconducting NbTi twisted pairs are used.^{2} Among these materials only Cu twisted pairs cause a significant passive load comparable to the one of the coaxial cables, see Fig. 1.
2.2 Active load
The active load in the dilution refrigerator depends on the level of attenuation of the RF lines and the installed attenuators, and on the signal levels required at the chip. Therefore, in this section, we briefly discuss how much attenuation is needed to reduce thermal noise and how to distribute the attenuators among the various temperature stages. Furthermore, we discuss the signal levels required in typical experiments with superconducting qubits.
2.2.1 The need of attenuation
For the discussion presented here we consider the signal frequency at \(\bar{\omega}/2\pi=6\mbox{ GHz}\) since typical qubits are operated at 5 GHz and typical readout resonators at 7 GHz. A lower bound for the total attenuation needed to achieve a noise photon number of \(n_{ \mathrm{MXC}}=10^{-3}\) is obtained by neglecting the second term on the r.h.s. of Eq. (3) yielding \(n_{\mathrm{BE}}(T=300 \mbox{ K},\bar{\omega})/10^{-3}=60\mbox{ dB}\). This is a lower bound since blackbody radiation emitted by attenuators at all other temperature stages is neglected.
We observe a similar feature when plotting \(n_{\mathrm{MXC}}\) as a function of \(A_{\mathrm{CP}}\), for a fixed attenuation of 20 dB at the 4 K and MXC stages respectively (green dashed line in Fig. 2). Here a little bit more than 20 dB attenuation would further reduce \(n_{\mathrm{MXC}}\), as expected from a reference value \(A_{\mathrm{CP, ref}}=n_{\mathrm{BE}}(T_{ \text{4 K}},\bar{\omega})/n_{\mathrm{BE}}(T_{\mathrm{CP}},\bar{ \omega})=22\mbox{ dB}\). We finally plot \(n_{\mathrm{MXC}}\) as a function of attenuation at MXC (red dash-dotted line in Fig. 2). It demonstrates that, over the considered attenuation range, adding more attenuation at MXC linearly decreases \(n_{\mathrm{MXC}}\). The reason is that the noise floor of thermal photons at MXC is as low as \(n_{\mathrm{BE}}(T_{\mathrm{MXC}}=20\mbox{ mK},\bar{\omega})=6\times10^{-7}\). However, we will see in Sect. 3.2 that the limited cooling power at MXC prevents us from installing significantly more than 20 dB of attenuation at MXC.
2.2.2 Signal levels required for the operation of the quantum processor
In this section we estimate the powers required at the chip for driving a π-pulse on a qubit and setting a flux bias on a qubit, two important operations on superconducting qubits.
To drive a π-pulse on a qubit, we apply an RF pulse at the qubit frequency with a Gaussian envelope through a CPW transmission line weakly capacitively coupled to the qubit. This coupling has to be sufficiently small to prevent Purcell decay of the qubit into the transmission line [39]. The finite value of the coupling imposes a limit on the \(T_{1}\) time of the qubit, which we target here to be no lower than \(T_{1}^{\mathrm{lim}}=2\mbox{ ms}\), corresponding to a reduction of the \(T_{1}\) time of a qubit with an intrinsic life time of 100 μs [5] by no more than 5%. The amplitude of the time-dependent Rabi frequency \(\varOmega_{R}(t) = \varOmega_{0} {\mathrm{exp}} [-t^{2}/(2\sigma^{2}) ]\) for a 20 ns long π-pulse [40] with a width \(\sigma=3.3\mbox{ ns}\) is given by \(\varOmega_{0}=\sqrt{\pi/(2\sigma^{2})}\approx2\pi\times60\mbox{ MHz}\). To achieve this value a peak power \(P_{\mathrm{p}}=\hbar\omega_{q} T _{1}^{\mathrm{lim}}\varOmega_{0}^{2}/4\approx-66\) dBm is needed [39, 41]. For an estimate of the associated heat load we employ the average power of the pulse, \(P_{\mathrm{avg}}=P_{\mathrm{p}}\frac{1}{6\sigma}\int_{-3\sigma} ^{3\sigma}{\mathrm{exp}} [-t^{2}/\sigma^{2} ]\,\mathrm {d}t=\frac{\sqrt{ \pi}}{6}P_{\mathrm{p}}\approx-71\mbox{ dBm}\). This number will be further reduced due to a finite duty cycle of those pulses during the execution of a quantum algorithm. We assume a maximum duty cycle of 33%, corresponding to an operation mode of the quantum processor where single- and two-qubit gates are alternated with the two-qubit gate duration being twice as long as the single-qubit gate duration [22, 42]. Also \(\pi/2\)-pulses require only a quarter of the power. Assuming an equal share between π- and \(\pi/2\)-pulses, we use an average required power per qubit drive line of −78 dBm for the estimates presented in Sect. 3.2. We note that the duty cycle of drive pulses can be significantly lower than 33% if the durations of two-qubit gates and readout pulses are significantly longer than the duration of drive pulses. A further reduction of the duty cycle arises if the repetition period of the algorithm is dominated by a wait time to reset the qubits to the ground state.
Readout signals used to drive readout resonators to infer the state of qubits are typically an order of magnitude smaller and have a lower duty cycle. They are therefore not taken into account in our estimates.
Concerning dissipation in flux lines, we primarily consider DC biasing currents, which are constantly applied to set the qubit frequency, in most of the cases to the so-called sweet spot \(\omega_{\mathrm{q}}= \omega_{0}\), at which the qubit frequency \(\omega_{q}\) is to first-order insensitive to flux noise [16]. Assuming a worst case scenario of random offset magnetic fields in the SQUID loops, the compensating flux \(\varPhi_{\mathrm{offset}}\) which we apply to reach the closest sweet spot, is equally distributed in the interval \([-\varPhi_{0}/2,\varPhi_{0}/2]\). This corresponds to a current interval of \([-1,1]\mbox{ mA}\), when using a reasonable mutual inductance of \(M=\partial\varPhi/\partial I=0.5 \varPhi_{0}/{\mathrm{mA}}\) between the flux line and the SQUID loop. The mutual inductance is determined by the coupling strength (proximity) of the on-chip flux line to the SQUID loop and the area of the SQUID loop. A large M on the one hand minimizes the required current and thus dissipation in the line, and allows for the use of low-noise current sources which typically have a small dynamic range. On the other hand, a large SQUID loop is also more susceptible to (external) magnetic flux noise, and care needs to be taken in the design of large coupling strengths in order to keep residual capacitive coupling of the qubit to the flux line low, which otherwise can limit the qubit lifetime due to the Purcell effect [16]. The dissipated heat due to the bias currents is most critical at the MXC stage which has the lowest cooling power. The amount of dissipation depends on the DC resistances of the stainless steel cable and low-pass filter installed between CP and MXC, and on the stage to which the dissipated heat predominantly flows to. We discuss the experimental determination of this heat load in Sect. 5.2.
We also consider dissipation due to flux pulses, which are used for the realization of two-qubit gates. Assuming that during the flux pulse the qubit frequency is tuned by about 10% of its sweet spot value [21, 22], a flux amplitude of \(\pm0.2 \varPhi_{0}\) is needed, corresponding to a current amplitude of 0.4 mA. The associated active loads are estimated in Sect. 5.2 using the results from the DC measurements. To estimate the duty cycle of flux pulses we consider alternating single- and two-qubit gates, as discussed above, and flux pulses applied only to one of the qubits during a two-qubit gate, yielding a duty cycle of \(0.5\times66\% = 33\%\).
2.3 Radiative load
Each temperature stage i except for the cold plate is fitted with a dedicated heat shield to protect the next lower temperature stage \(i+1\) from radiative load from temperature stage \(i-1\). For a given temperature stage i we calculate the radiative heat load on heat shield i from the enclosing heat shield by assuming infinitely extended cylindrical heat shields, and solving a system of coupled heat equations. The heat shields at the 50 K and 4 K stages are made of Aluminium, and of Cu on Still and MXC. They are characterized by their emissivity \(\epsilon=0.06\), as quoted by the manufacturer. Compared to the cooling power of each stage, we find a significant contribution only for the 50 K stage, amounting to ∼50 W. This corresponds to about half of the nominal cooling power of the two pulse tube coolers (2x Cryomech PT 420). This load however has already been taken into account in the available cooling power as specified by the manufacturer (40–50 W). Radiative loads at the lower temperature stages are insignificant [32].
3 Cryogenic setup
3.1 Dilution refrigerator (DR)
Furthermore, the distance between the Still plate and the CP has been enlarged by 60 mm to provide more space for the inline integration of MW components, and for reducing the passive heat load due to the cables from Still to CP. The dilution refrigerator was pre-equipped by the manufacturer according to our specifications with two looms of Cu twisted pairs (AWG 35) and two looms of Ph-Br twisted pairs (AWG 36), running both from room temperature to 4 K. From 4 K to MXC two looms of NbTi twisted pairs were pre-installed. Each of the looms contains 12 twisted pairs.
We have measured the available cooling power on each stage of our custom dilution refrigerator (in the presence of the pre-installed DC wiring), see Table 1, using the technique described in Sect. 4.1. These powers are not to be exceeded to guarantee a base temperature below 20 mK [43]. We designed and planned the wiring and attenuation scheme of the dilution refrigerator with the goal that the resulting heat load is at least about a factor three smaller than those cooling powers. The table also contains the RF cable lengths between the stages. Note that the cables have bends (bending radius ∼1 cm) to reduce strain on RF connectors when the system is cooled down.
As discussed in Sect. 2, good thermalization of the cables and attenuators is important for both reducing thermal noise photons affecting qubit coherence and reducing the heat load on the lower temperature stages affecting the cryostat performance. For this purpose, we have developed a common thermalization method for both attenuators and cables. It consists of a Cu plate with hexagonal cutouts, in which we individually clamp attenuators using a wedge shaped Cu piece, which presses the attenuator against two faces of the cutout when tightening a three millimeter (M3) Brass screw (Fig. 4(a) and (b)). We apply a torque of 0.9 Nm on each screw for fastening. We use Brass screws as they thermally contract more than Cu when cooled down, providing additional clamping force to the attenuator. This is important, because in general the contact resistance between two metal pieces at low temperatures is dominated by clamping force rather than surface area [44]. In addition to thermalizing attenuators we thermalize the outer conductors of RF cables at stages without attenuators using the same Cu plate (Fig. 4(c)) in combination with a two-part Cu adapter (Fig. 4(d)). The round flange on those pieces guarantees radiation tight mounting. Using those plates we fit 25 RF lines into each of the six LOS ports.
We now continue to describe in more detail the custom cabling of the dilution refrigerator. Schematics of drive lines, flux lines, and lines for readout are shown in Fig. 3 b and are typical for superconducting qubit experiments. To minimize passive heat load, we choose UT-085-SS-SS cables for all types of lines except for the output lines.^{3} We characterize and test the system at half of its capacity by installing three out of six possible cable trees in the dilution refrigerator: a cable tree comprising of 25 drive lines, a cable tree comprising of 25 flux lines, and a cable tree comprising of lines needed for readout. This readout cable tree consists of six drive lines and 5 pump lines (see Sect. 3.4), running both from top to bottom of the dilution refrigerator, and four output lines running from room temperature to 4 K. The remaining parts of the output lines continue in the non-LOS cutouts below one of the pulse tube coolers, where more space is available for larger components of the output lines.
3.2 Drive line configuration
For simplicity we stick to the values for which attenuation in cables has been neglected in the following discussion of \(n_{\mathrm{MXC}}\) (solid bars in Fig. 5(a)). We first consider configuration \(\mathrm{C}1=\{0, 10, 0, 20, 30\}\mbox{ dB}\), for which we calculate \(n_{\mathrm{MXC}}\approx0.0012\). This is only slightly larger than the lower bound \(n_{\mathrm{MXC,min}}\approx0.001\) for a total of 60 dB of attenuation, corresponding to a configuration \(\mathrm{C}_{\min}=\{0, 0, 0, 0, 60\}\mbox{ dB}\), see also Sect. 2.2.1. The reason for the low \(n_{\mathrm{MXC}}\) of C1 is that \(n_{\mathrm{MXC}}\) considered as a function of attenuation on each of the stages is in the linear regime for configuration C1, see Fig. 2(a).
While the relative loads on the 50 K, 4 K, and Still stages are negligibly small, the CP and MXC stages are subject to a significant load.
Since configuration C1 has a particularly large relative load of 35% on the CP stage, we consider configurations with a total of 40 dB attenuation on the CP and MXC stages instead of 50 dB. In particular, we target configurations with relative loads below 5%. This is to keep some margin, e.g. for the installation of absorptive infra-red blocking filters, see end of this Section. Removing 10 dB of attenuation at the CP and adding 10 dB at the 4 K stage instead results in \(\mbox{C}2=\{0, 20, 0, 10, 30\}\mbox{ dB}\) and reduces the active load on the CP by a factor 10. The resulting value of \(n_{\mathrm{MXC}}=0.002\) is almost a factor of two larger than in C1 because 20 dB of attenuation at the 4 K stage already corresponds to the regime where \(n_{\mathrm{MXC}}\) as a function of \(A_{\text{4 K}}\) starts to saturate to the 4 K noise floor, see Fig. 2(a). We also note that dissipation on the 4 K stage is relatively low compared to the available cooling power and thus redistributing some of the attenuation at 4 K to the 50 K stage is not necessary. Configuration \(\mbox{C}3=\{0, 20, 0, 20, 20\}\mbox{ dB}\) corresponds to C2 with 10 dB of attenuation moved from the MXC to the CP stage. In this case the noise photon number only increases by 20% compared to C2 to \(n_{\mathrm{MXC}}=0.0024\). This is because at \(A_{\mathrm{CP}}=20 \mbox{ dB}\) \(n_{\mathrm{MXC}}\) considered as a function of \(A_{\mathrm{CP}}\) is still relatively far from reaching the CP noise floor. At the same time the relative active load on the MXC stage is reduced by a factor of 10 to <1%.
To further reduce the active load on the CP by a factor ten one can redistribute 10 dB from the CP to the Still, resulting in configuration \(\mbox{C}4=\{0, 20, 10, 10, 20\}\mbox{ dB}\). However, \(n_{\mathrm{MXC}}\) thereby increases by a factor of two compared to C3, which can be explained with similar arguments as above. A relatively small increase in \(n_{ \mathrm{MXC}}\) as compared to C3 can be achieved by redistributing only 3 dB from the CP to the Still, reducing the load on the CP by a factor two. However, this is not very practical since it comes along with a non-standard attenuation of 17 dB at the CP. In addition, solutions with four attenuators require an additional hardware cost of one attenuator and one connectorized RF cable per line as compared to solutions with only three attenuators per line.
As a compromise between a low number of noise photons and a low heat load on MXC we select C3, with a noise photon number of about 0.001 (taking attenuation in cables into account).
To filter infrared radiation that can lead to quasi-particle generation and thus to a reduced \(T_{1}\) time [8–13], we provide the option to install a custom-built, IR filter based on Eccosorb CR-124 absorber material right before the attenuator at MXC, see Appendix 3. With an attenuation of 6 dB at 6 GHz, it further reduces \(n_{\mathrm{MXC}}\) by a factor four, while increasing dissipation on all stages by a factor four due to the larger power required at the input of the cryostat to achieve the same signal level at MXC.
3.3 Flux line configuration
Since the bandwidth of typical flux pulses is at most 1 GHz [21, 22, 42], we install a low-pass filter with a 3 dB bandwith of about 1.9 GHz (Minicircuits Minicircuits VLFX1300+) in the flux line to suppress thermal noise photons at qubit frequencies, which could otherwise couple to the qubit due to parasitic capacitive or inductive coupling of the on-chip flux line and the qubit. A custom-built, absorptive IR filter is installed with the goal to reduce quasi-particle generation in the superconducting film, see above and Appendix 3 for details. It is mounted in the RF lines at the MXC stage to achieve the lowest infrared noise photon number.
3.4 Output line configuration
A reflective bandpass filter (Keenlion KBF-4/8-2S) at MXC suppresses noise at frequencies outside the bandwidths of circulator and isolators, see Fig. 6(c). To minimize signal loss between the two amplfiers, we use superconducting NbTi coaxial cables between MXC and 4 K. To minimize passive heat load we chose stainless steel cables from 4 K to room temperature. According to Friis formula the increased loss compared to e.g. SPCW cable does not decrease the signal to noise ratio of the amplifier chain. E.g. a loss of 10 dB in the cable section between HEMT and the first amplifier at room temperature can be tolerated if one uses a low noise amplifier with a noise temperature of about 500 K as a first amplifier at room temperature.^{4}
For the operation of the TWPA a microwave pump signal with a typical frequency in the range 6–8 GHz and a power level at the input of the TWPA of about −55 dBm is required. This pump signal is added to the TWPA through the isolated port of a 20 dB directional coupler^{5} (Krytar 120420), see Fig. 3(b). To realize the relatively large pump power we use pump lines with a reduced total attenuation of 50 dB attenuation as compared to the 60 dB attenuation used in the drive lines. This 50 dB of attenuation includes the 20 dB attenuation provided by the directional coupler with its coupled port terminated with a \(50~\varOmega\) load resistance. Photographs of the readout components described in this Section are shown in Fig. 6. The TWPAs and the isolator arrays are mounted on 2 mm thick Cu sheets whose stability is enforced by Cu bars at their edges, see Fig. 6(a). For compact mounting and thermalization, the four TWPAs and the four bandpass filters (Fig. 6(c)) are first stacked using Cu spacers, which are then mounted to the Cu sheet or to an L-shaped bracket, respectively. Similarly, the four HEMT amplifiers and the four circulators are first mounted onto Cu pieces which are then mounted to a common bracket.
4 Direct measurements of the passive heat load
To determine the passive heat load induced by each cable tree and to extract a heat load per line, we performed reference measurements of the cooling power in the dilution refrigerator system as delivered without microwave cabling. Then we cooled down the system each time we installed a new cable tree and recorded the temperature increase on each of the stages of the dilution refrigerator to infer the corresponding passive heat loads. The cooldown of cable trees comprising up to 25 identical lines of a given type has the advantage of generating temperature increases which are larger than those created by a single line and also larger than typical run-to-run temperature variations on the stages of the dilution refrigerator. Furthermore, this method naturally averages over variations in material and components, as well as over possible variations occurring in the installation and thermalization of individual cables and attenuators.
4.1 Reference measurements
The cooling power on the Still stage is provided mainly by the evaporation of ^{3}He and reflects the dependence of the partial pressure of ^{3}He on temperature [51, 52]. The temperature of the Still plate increases by about 60% over the range of applied heat powers of 0–18 mW, see green data set in Fig. 7(c). Importantly, the heat power applied to the Still stage also sets the cooling power on MXC, \(P_{\mathrm{MXC}}\propto\dot{n}_{3}T_{ \mathrm{MXC}}^{2}\) [51] because it regulates the flow \(\dot{n}_{3}\) of the ^{3}He through the dilution unit and thus across the phase boundary between the concentrated and dilute phases of ^{3}He in the mixing chamber. Since an increased Still temperature also creates an additional load on the CP and MXC stages, there is an optimum flow value which maximizes the cooling power for a given \(T_{\mathrm{MXC}}\). The \(T_{\mathrm{MXC}}^{2}\) dependence of \(P_{\mathrm{MXC}}\) arises because the enthalpy difference between the two phases is proportional to \(T_{\mathrm{MXC}}^{2}\) [51]. For all measurements we set \(P_{\mathrm{Still}}=10.7\mbox{ mW}\), corresponding to a flow of 0.69(1) mmol/s and a cooling power on MXC of about 5 μW (13 μW) at 13 mK (20 mK), see Fig. 7(e). Importantly, when increasing the flow to 1.0 mmol/s by applying a heat power of 40 mW to the Still we reach a cooling power of 19 μW (540 μW) at 20 mK (100 mK).
The CP has no active cooling element, but since it is mounted on top of the last set of heat exchangers [52] it is effectively cooled by the cold mixture entering and leaving the mixing chamber. We measure the temperature increase over a relatively wide range of heat powers up to 300 μW since our attenuator configuration C3 in the drive lines is expected to create a significant active load on the CP. Over this range the temperature of the CP doubles from about 80 mK to about 160 mK, see turquoise data set in Fig. 7(d). The effect on \(T_{\mathrm{MXC}}\) is relatively small, e.g. at the largest measured value of \(P_{ \mathrm{CP}}=300~\mu\mbox{W}\) \(T_{\mathrm{MXC}}\) rises from 6.1 mK to 8.1 mK, corresponding to an effective load of only 1 μW on the MXC stage.
4.2 Determination of passive heat loads
Additional care needs to be taken when determining \(\Delta T_{ \mathrm{CP}}\) and \(\Delta T_{\mathrm{MXC}}\). This is because an increased Still temperature changes the flow through the dilution unit thereby not only leading to an additional passive load on CP and MXC, but effectively changing the cooling power vs. temperature curves. This is most evident for the MXC stage when considering the relation \(P_{\mathrm{MXC}}\propto\dot{n}_{3}T_{\mathrm{MXC}}^{2}\) discussed above. Before determining \(\Delta T_{\mathrm{CP}}\) and \(\Delta T_{ \mathrm{MXC}}\) we thus reduce the Still heat power such that we reach \(T_{\mathrm{Still}}=882(1) \mbox{ mK}\), resulting in the same flow of 0.69(1) mmol/s set for the reference measurements.
To extract the passive heat loads \(\Delta\tilde{P}_{i}\) for each type of line we cooled down the dilution refrigerator after individual installation of the drive line cable tree, the readout cable tree without the NbTi lines, the four NbTi output lines between the 4 K stage and the MXC stage, and the flux line cable tree, respectively. From the extracted total passive heat load after each cooldown we calculate an average load per line by dividing the total loads by the number of lines in the cable tree.
Passive heat load per line. Passive heat load (HL) of cable types installed in the DR, as inferred from observed temperature increases after the installation of individual cable trees into the DR. The upper and middle sections of the table refer to 0.085” diameter stainless steel coaxial cable (UT-085-SS-SS) with attenuator configurations used in drive lines and flux lines, respectively. Indicated errors include statistical errors between different cooldowns and reflect run-to-run temperature variations on the stages of the DR. The intervals of estimated heat loads correspond to calculations of lower and upper bounds (see text for details)
50 K | 4 K | Still | CP | MXC | |
---|---|---|---|---|---|
Drive line UT-085-SS-SS | |||||
Measured HL | 45(34) mW | 1.0(5) mW | 4(3) μW | 0.4(2) μW | 0.013(6) μW |
Estimated HL | 24–27 mW | 0.4–1.9 mW | 1.6–2.1 μW | 0.33–0.60 μW | 0.004 μW |
Flux line UT-085-SS-SS | |||||
Measured HL | 56(39) mW | 1.2(8) mW | 2(1) μW | 0.3(1) μW | 0.029(5) μW |
Estimated HL | 24–27 mW | 0.4–1.9 mW | 1.6–2.1 μW | 0.24–0.33 μW | 0.005–0.282 μW |
Output line UT-085-NbTi | |||||
Measured HL | – | – | – | 0.3(3) μW | 0.020(16) μW |
Estimated HL | – | – | – | 0.18–0.31 μW | 0.002–0.322 μW |
While the measured loads on the 50 K, 4 K, Still, CP stages agree within error bars with the estimated values, the load on the MXC stage is larger than predicted. This can have several reasons. First, there could be discrepancies between the thermal conductivitiy of the stainless steel (SS) of the cables and the data used for the estimates in that temperature interval. Second, thermalization of the center conductor via the attenuator at the CP could be incomplete. Third, the long timescales for thermalization of components at MXC, in particular if they contain hydrogen [53], can lead to an overestimation of the heat load since we measure \(\Delta T_{i}\) always at the same time (∼1 day after the condensation of the mixture when the MXC temperature is sufficiently stable). Fourth, gas desorbing from newly installed components may freeze out at MXC, creating an additional heat load.
The average passive loads per flux line on the 50 K, 4 K, and Still stages agree within error bars with the corresponding passive loads per drive line, see second section of Table 2. This is expected since the two types of lines are nominally identical from the vacuum flange to the Still stage. The load on the CP is comparable to the corresponding load per drive line. This observation is expected because the main contribution to this load originates from the outer conductor. The intervals for the estimated loads are calculated for complete thermalization of the center conductor and dielectric at the 50 K, Still and CP stages or no thermalization of the center conductor and dielectric on these stages at all. We exclude the case in which the center conductor and dielectric thermalize on the CP but not on the Still because the thermal contact resistance increases with decreasing temperature. The measured passive load of 0.029(5) μW per flux line on MXC is more than a factor of two larger than the passive load per drive line. We explain this observation by the absence of an attenuator in the flux line at the CP causing a direct passive load from the Still to the MXC stage via the center conductor and dielectric. The estimated upper bound of 0.282 μW is calculated by assuming no thermalization of the center conductor and dielectric on neither the CP nor the Still stage. Since this upper bound is by a factor of ten larger than the measured value, it is likely that the center conductor and dielectric thermalize at the Still stage. In this case the estimated load amounts to 0.054 μW.
The passive loads extracted from the cooldown of the four NbTi lines are listed in the third section of Table 2. These loads lie within the estimated lower and upper bounds corresponding to complete thermalization of the center conductor and dielectric at Still and CP or no thermalization at all, respectively. Using an argument analogous to the one used for the flux lines, it is likely that the center conductor and dielectric thermalize at the Still stage. How well the center conductor and dielectric thermalize at the CP cannot be evaluated given the size of the error bars of the extracted passive loads.
In summary, after the installation of 66 RF lines, mainly UT-085-SS-SS, \(T_{\mathrm{MXC}}\) increased from 6.1(1) mK to 8.4(2) mK, corresponding to a total passive heat load on MXC of 1.4(2) μW. This is more than a factor of ten smaller than the available cooling power at 20 mK, and thus sets the stage for experiments with tens to hundreds of superconducting qubits.
5 Active load measurements
To determine the active load during the operation of a quantum processor we apply continuous signals of a given power to a single drive line or current to a single flux line, and measure the resulting heat loads on each of the stages of the DR. Estimating typical duty cycles and power levels discussed in Sect. 2.2 allows us to estimate the total active load from the measured loads.
5.1 Drive lines
To experimentally test loads due to signal dissipation in drive lines we apply a RF signal to one of them and measure the corresponding temperature increases on the stages in the dilution refrigerator. Several drive lines were tested and their results agreed within error bars. In particular, we attempt to evaluate whether the power dissipated in an attenuator predominantly flows to the stage on which the attenuator is thermalized, or whether part of that heat flows to the next lower stage. This is important because the signal levels at stages with attenuators differ by 20 dB in the selected attenuator configuration of \(\mbox{C}3=\{0, 20, 0, 20, 20\}\mbox{ dB}\). In other words, the power dissipated in an attenuator at the 4 K (CP) stage is by a factor 100 larger than the power dissipated in the attenuator at the CP (MXC) stage. Hence, a small fraction of heat dissipated in an attenuator at a given stage flowing towards the lower temperature stages can significantly increase the heat load on those stages.
We first apply a power of 10 dBm at a frequency of 10 MHz, at which dissipation in the RF cables themselves is negligible. This allows for a simple comparison with the expected loads of 10 mW on the 4 K stage, 100 μW on CP, and 1 μW on MXC. Using our reference measurements, we extract an active load of 11.5(1.6) mW on the 4 K stage, 95(9) μW on CP, 1.1(1) μW on MXC, in agreement with the predicted values. On the Still stage we measure an active load of 48(6) μW, which we attribute to a small fraction of power dissipated at 4 K (about 0.4%) flowing towards the Still stage. Our measurements show that more than 99% of the power dissipated in an attenuator flows to the stage on which the attenuator is anchored, thereby validating our method for thermalizing attenuators. We note that the extracted active loads represent pure active loads, i.e. we subtract from the measured load on a given stage the additional passive load due to an increased temperature of the next higher stage.
We also measured the active load per TWPA pump line and found values comparable to the active load per drive line. This results from the fact that the required pump power at the input of the TWPA is with about −65 dB by 13 dB larger than the required average power in a drive line but the TWPA pump lines having 10 dB less attenuation than the drive lines, see Fig. 3(b). The largest relative active load of a pump line arises at the CP stage and amounts to 0.1%, corresponding to 0.02 μW in absolute numbers.
In this section, we discussed the experimentally determined active loads for the selected attenuator configuration {0, 20, 0, 20, 20} dB in the drive lines. Since this configuration has been chosen to keep the active load significantly below the available cooling powers, the active load due to execution of single-qubit gates has only a small influence on the temperature of the quantum processor. Whether the thermal photon number is as low as predicted needs to be experimentally determined [3, 4] and would provide information about the effective temperatures at which the lowest temperature attenuators emit blackbody radiation.
5.2 Flux lines
For a reasonable choice of mutual inductance between flux line and qubit discussed in Sect. 2.2.2, the DC biasing currents needed to tune the qubits to the desired transition frequencies are considered to be randomly distributed in the interval \([-I_{ \mathrm{max}},I_{\mathrm{max}}]=[-1,1]\mbox{ mA}\). Hence, an average power of \(P_{\mathrm{avg}}=(1/I_{\mathrm{max}})\int_{0}^{I_{\mathrm{max}}} R _{\mathrm{eff}} I_{\mathrm{MXC}}^{2}\mathrm{d}I_{\mathrm{MXC}} = \frac{1}{3} R_{\mathrm{eff}} I_{\mathrm{max}}^{2}\) is dissipated, evaluating to 0.050(3) μW (0.140(7) μW) on MXC (CP). For 25 flux lines this sums up to 1.25(7) μW (3.5(2) μW) on MXC (CP), which is about a factor of 15 (40) smaller than the cooling power available at MXC (CP) at a temperature of 20 mK (140 mK). We note that this is a worst-case scenario since with careful magnetic shielding it should be possible to reduce the offset fluxes to values close to zero.
6 Conclusion and outlook
Based on the presented measurements of passive and active loads we estimate the total heat load acting on the different stages of a dilution refrigerator, see Fig. 10, when operating a 50 qubit processor with individual drive and flux control and with a multiplexed readout architecture allowing for simultaneous readout of sets of 6–7 qubits. Such a quantum processor requires a total of 124 RF lines (50 drive lines, 50 flux lines, 8 output lines, 8 readout resonator drive lines, 8 TWPA pump lines), corresponding to the operation of the presented system at its full capacity. The largest relative loads of about 30% occur at the CP and MXC stages, with about an equal share between passive and active loads. The load on MXC corresponds to an operation temperature of 14 mK. The active load in the drive lines is a result of targeting a noise photon number of 10^{−3} or less. The relative loads of about 20% and 10% on the 50 K and 4 K stages, respectively, are dominated by passive loads. The load on the Still stage is negligibly small. Hence, the thermal performance of the discussed system would allow for the operation of a 150 qubit processor if the capacity for hosting coaxial lines was increased three-fold. This capacity increase could be achieved either by increasing the density of coaxial line integration or by building a larger diameter dilution refrigerator.
The material of the coaxial cables has been chosen to minimize passive heat load on all of the stages. However, instead of the standard 0.085” diameter coaxial cables, one could use 0.047” cables or even thinner ones. Using smaller diameter cables is beneficial because the passive heat load scales with the square of the diameter, whereas the attenuation in the cable scales approximately linearly. For the proposed 0.047” SS-SS cables, the passive heat load would be reduced by more than a factor three. Since large attenuation is desired in the drive lines, the active load due to the execution of single-qubit gates would not be increased if one reduced the attenuation of the installed attenuators accordingly. However, the dominant active load at MXC due to flux biasing of the qubits would be increased more than three-fold because the DC resistance of the cables scales inversely with their diameter. The active load in the flux lines could be brought to close to zero though when using superconducting NbTi cables.
As an outlook, we briefly discuss an optimistic scenario in which magnetic flux offsets are brought close to zero by careful magnetic shielding of the quantum processor. Thus avoiding the active load due to DC flux biasing and considering the use of 0.047” diameter cables, the dominant relative load on the CP stage amounts to about 14%, allowing for the operation of about \(50/0.13\approx360\) qubits.
Ultimately, we propose the following modifications to allow for the operation of about one thousand qubits. First, we use superconducting NbTi cables for the flux lines eliminating active load in these lines. Second, we tolerate a CP temperature of about 200 mK at which a cooling power of at least 400 μW is available. To maintain the noise photon number below 10^{−3} we shift about 7 dB of attenuation from the CP to the MXC stage, increasing the active load on MXC by a factor six. Third, we tolerate a base plate temperature of 30 mK at which the cooling power at MXC is at least doubled to about 40 μW, while not affecting the noise photon number. We note that in this scenario active loads in the drive lines are the dominating heat load at CP and MXC.
We conclude with the remark that future large-scale dilution refrigerators will likely have several dilution units and/or larger ones to provide sufficient cooling power at the CP and MXC stages.
Silver plated copper weld, also referred to as silver plated copper clad steel per ASTM B-501. It is composed of a steel core and a copper cladding. For our estimate, we model it as a single core Cu wire with an RRR of 100.
For clarity, NbTi twisted pairs are considered to consist of pure NbTi, although they are typically embedded in a CuNi matrix for mechanical stability.
Smaller diameter cables of the same type would further reduce the passive heat load, but introduce a larger frequency dependence in the S21 transmission parameter.
When using only a HEMT amplifier as a cryogenic amplifier (and not a TWPA), an ultra-low noise amplifier with a noise temperature of about 100 K is recommended.
Choosing a smaller coupling ratio would decrease the transmission of the readout signal through the directional coupler accordingly.
We assume the dielectric (Teflon) to be thermalized with the center conductor because the thermal expansion coefficient of Teflon is a factor of six larger than the one of stainless steel.
Declarations
Acknowledgements
The authors thank L. DiCarlo, Y. Salathé, G. Norris, C. K. Andersen, and S. Gasparinetti for useful discussions, and M. Frey for support during the installation of the dilution refrigerator.
Availability of data and materials
The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.
Funding
This work is supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via the U.S. Army Research Office grant W911NF-16-1-0071 and by ETH Zurich. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the ODNI, IARPA, or the U.S. Government.
Authors’ contributions
The design of the cryogenic setup was developed by SK, PK, RK, JL, CE and AW. The setup was built by SK, SS and JH. The experiments were performed by SK and SS. The data was analyzed and interpreted by SK, SS, PK and PM. The manuscript was written by SK, SS, CE and AW. The project was led by AW. All authors commented on the manuscript. All authors have read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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