## Improving Superconducting Qubits

### Understanding Decoherence

- Fig. 1: (a) diagram of an ideal circuit comprising an inductor L and a capacitor C (top) and its evenly spaced quantum energy levels in the harmonic oscillator potential (bottom). (b) circuit that includes a Josephson junction Jj (top) and its energy levels in an anharmonic potential (bottom). The two lowest levels are the ground (g) and excited (e) qubit states.
- Fig. 2: (a) the electron and hole components of a quasiparticle acquire opposite phases ϕ when tunneling thorough a Josephson junction and can therefore interfere destructively. (b) T1 time as a function of external flux Φext in a fluxonium qubit. Changing the flux modifies the phase difference through the junction. The peak in T1 is due to the destructive interference. The fluctuations are caused by changes in the number of quasiparticles (proportional to xqp) between repeated experiments. Adapted from [6].
- Fig. 3: (a) the sequence of pulses used to excite the qubit repeatedly, before a final measurement pulse (b) excited state probability vs time for different numbers of pumping pulses. The more pulses are applied, the longer is the qubit lifetime, defined as the time it takes for the signal to decay by a factor of 1/e. Adapted from [9].

**The idea of a quantum computer can be traced back to Feynman’s talk at the 1981 conference on “The physics of quantum computation”: he recognized that simulation of nature should be quantum mechanical, “a wonderful problem, because it doesn’t look so easy” [1]. Indeed, building a quantum computer is not easy: it requires the so-called DiVincenzo criteria to be met [2]. In particular, the lifetime of a quantum bit (qubit) should be long enough, so that errors can be corrected and a useful computation performed.**

Various possibilities to realize a physical qubit are being explored, from natural systems, such as photons and ions, to engineered ones, like quantum dots and superconducting circuits. The latter have been made exponentially better since the first demonstration in 1999 [3]. The understanding of the physical processes limiting the lifetime contributes to our knowledge of fundamental aspects of superconductivity on one hand, while it can lead to further improvements of qubits on the other. One such improvement relies on a new technique to dynamically reduce noise in a stochastic way.

**Superconducting Circuits**

Electrical circuits are formed by combining three basics elements: capacitors, inductors, and resistors. The latter are responsible for losses: an ideal circuit with only capacitors and inductors would oscillate indefinitely, but resistance is always present and dampens the oscillations. If inductors and capacitors are made of a superconductor the losses are extremely small, the oscillations can behave quantum mechanically and the circuit displays a discrete set of energies, see Fig 1(a). However, such a quantum harmonic oscillator is not a qubit: the energies are equally spaced, so the corresponding quantum states cannot be manipulated individually. To solve this problem, an additional circuit element is needed, a Josephson junction, which can be formed by introducing a thin insulating barrier between two superconducting electrodes. Circuits with Josephson junctions have unequal energy spacing and can be employed as qubits, see Fig 1(b). Various superconducting qubits (Cooper pair box, transmon, fluxonium, …) differ in the number of basic elements and in the way they are combined, but share the same physical platform.

**Decoherence and Quasiparticles**

While the losses in a superconducting circuit are small, they nonetheless limit the lifetime of a qubit.

There are two timescales that characterize the qubit: the relaxation time T_{1} and the decoherence time T_{2}, a terminology borrowed from NMR. They give the inverse rate of exponential decay of a quantum state. For example, if the qubit is in the excited state at time t=0, the probability to find it in that state at a later time t is P(t)=exp[-t/T_{1}]. In other words, after a time longer than T_{1 }there is high probability that the qubit has relaxed to the ground state. For quantum information applications, the time T_{2} is more important as it denotes the lifetime of a quantum superposition of the two qubit states. It is always smaller than twice T_{1}, but it can be shorter than this upper limit if so-called pure dephasing processes are present. Currently, the best superconducting qubits have lifetimes that range from tens of microseconds to about a millisecond. These lifetimes are much longer than the time it takes to perform operations (few tens to few hundreds of nanoseconds); this has already enabled some partial error correction and basic quantum simulation to be demonstrated.

A process that limits the qubit lifetime is due to its interaction with quasiparticles. In a superconductor, electrons pair up to form Cooper pairs, which can flow without resistance. However, at any finite temperature (or because of some external disturbance) not all the electrons are paired – this is the origin of the small residual losses. The excitations in the superconductor due to unpaired electrons are more appropriately called quasiparticles, since they are a coherent superposition of electron- and hole-like excitations and do not have a definite charge.

**Measuring Quasiparticles**

According to theoretical predictions, the relaxation time of a qubit is inversely proportional to the number of quasiparticles that attempt to tunnel through a Josephson junction [4]. By increasing the temperature, more Cooper pairs are broken, and the expected shortening of T_{1} has been measured in a transmon qubit [5]. The fact that quasiparticles are a coherent superposition of electrons and holes leads to the possibility of destructive interference between the two components, Fig. 2(a). This interference in turn causes an increase of T_{1}, which has been measured with a fluxonium qubit [6], see Fig. 2(b). Remarkably, such an interference effect was predicted in the ‘60s by Nobel-laureate Brian Josephson, but his prediction had to wait 5 decades to find experimental verification.

These works provide convincing evidence of the detrimental effect of quasiparticle on the qubits. We can exploit the sensitivity to quasiparticles of the qubit to use the latter as a detector. By measuring how the qubit lifetime changes after adding on purpose quasiparticles, we can study the dynamics of the latter, enabling for example the estimation of their diffusion constant. Using this approach, it was revealed that superconducting vortices can capture quasiparticles in their cores [7], resembling what happens to a floating body near a whirlpool. In this way, vortices speed up the return of the qubit to its original condition: T_{1} is short initially, when there are many quasiparticles, and becomes long again after the quasiparticles have been captured.

**Controlling Quasiparticles**

The speed-up afforded by vortices hints to the possibility that controlling quasiparticles can improve the qubit properties, but using vortices is not practical. A more reliable way to capture quasiparticles is achieved by using small normal-metal islands as traps [8]. Experiments show that by increasing the trap length from tens to hundreds of micrometers, the rate at which quasiparticles are absorbed increases, up to a rate limited by the inverse of the diffusion time of the quasiparticles in the superconductor; this finding is in agreement with theoretical expectations.

The normal-metal traps have been demonstrated in relatively large transmon qubits, with sizes of a fraction of a millimeter, but traps are more difficult to engineer in devices such as the flux qubit, which has small superconducting islands of micrometer size. Using such a device [9], a new technique to limit the quasiparticle influence has been devised which resemble some well-known techniques in NMR. For example, by applying so-called refocusing pulses (Hahn spin-echo, CPMG sequence, etc.) the pure dephasing due to low-frequency noise can be largely suppressed and hence T_{2} increased. In another approach, dynamical nuclear polarization (DNP), the polarization of electrons is transferred to nuclei to boost the signal from the latter. In quantum dot-based qubits, the larger nuclear spin polarization also increases T_{2}.

The new technique is inspired by both pulse sequences and DNP, but leads to an increase of T_{1}. It works as follows: a sequence of pulses is used to bring the qubit into the excited state, with the pulse spacing ∆T comparable to T_{1}, see Fig. 3(a). This means that with high probability the qubit decays between pulses. In decaying, the qubit gives its energy to a quasiparticle. A higher-energy quasiparticle diffuses faster, which helps removing the quasiparticles from the qubit. The process is stochastic, since the qubits can relax via other mechanisms, without giving its energy to a quasiparticle, and the quasiparticle does not always leave the qubit. However, by repeating this pumping process many times, quasiparticles can be expelled and their number reduced. Then the measured lifetime at the end of the sequence is longer by a factor of 3, Fig.3(b). A drawback is that quasiparticles will eventually return, but combining the pumping technique with traps could prolong its effectiveness. Essentially, the approach works by shaping the noisy environment of the qubit, and could perhaps be applied in other contexts with appropriate modifications.

**Author**

Gianluigi Catelani

**Contact
Dr. Gianluigi Catelani**

Peter Grünberg Institut (PGI-2)

Forschungszentrum Jülich

Jülich, Germany

g.catelani@fz-juelich.de

References

[1] R. P. Feynman, Int. J. Theor. Phys. 21, 467 (1982)

[2] D. P. DiVincenzo, Fortschr. Phys. 48, 771 (2000)

[3] M. H. Devoret and R. J. Schoelkopf, Science 339, 1169 (2013)

[4] G. Catelani et al., Phys. Rev. Lett. 106, 077002 (2011)

[5] H. Paik et al., Phys. Rev. Lett. 107, 240501 (2011)

[6] I. Pop et al., Nature 508, 369 (2014)

[7] C. Wang et al., Nat. Commun. 5, 5826 (2014)

[8] R.-P. Riwar et al., Phys. Rev. B 94, 104516 (2016)

[9] S. Gustavsson et al., Science 354, 1573 (2016)