Noise Models Across Quantum Hardware Modalities

Introduction

In this blog post I want to walk you through how I think about noise in real quantum hardware, across several of the leading modalities: superconducting circuits, trapped ions, neutral atoms, spin qubits, photonic qubits and topological qubits based on Majorana zero modes. Modern devices are already incredibly sophisticated, yet all of them are still noisy, and understanding that noise in a quantitative way is essential if we want to scale to fault tolerant quantum computing.

Across platforms, I find it useful to keep a very simple, unifying picture in mind. A qubit is an effective two level system with Hamiltonian

where \(\omega_0\) is the qubit transition frequency. In reality, this ideal Hamiltonian is perturbed by time dependent couplings to many uncontrolled degrees of freedom. I can write that schematically as

where \(H_{\text{control}}(t)\) contains the carefully engineered microwave, laser or electrical control fields, and \(H_{\text{noise}}(t)\) captures unwanted fluctuations from the environment and from imperfect control electronics.

From this starting point, I almost always classify noise processes into relaxation, which changes the qubit population and is characterised by a time scale \(T_1\), and dephasing, which randomises the phase of superpositions and is characterised by a time scale \(T_2\) or \(T_2^*\). A very standard relation that you will see again and again is

where \(\Gamma_\phi\) is the pure dephasing rate. All of the detailed physics of a specific platform is hidden in how \(T_1\) and \(\Gamma_\phi\) depend on materials, geometry and control.

In what follows I will go modality by modality and build up simple but reasonably realistic noise models, with a few equations to make the connection between microscopic fluctuations and effective decoherence rates.

Superconducting qubits

Superconducting qubits are anharmonic oscillators built from Josephson junctions and linear inductors and capacitors. Their effective Hamiltonian is often written as a weakly anharmonic oscillator with a qubit subspace defined by two lowest energy eigenstates. From a noise perspective it is convenient to express the qubit Hamiltonian as

where \(\omega_q\) is the mean qubit frequency and \(\delta \omega_q(t)\) are frequency fluctuations coming from charge noise, flux noise, critical current noise and photon shot noise in the readout resonator [1][2][3].

Energy relaxation \(T_1\)

For superconducting qubits, \(T_1\) is often limited by dielectric loss, Purcell decay through the readout resonator, and quasiparticle tunnelling across Josephson junctions [1]. In a Golden rule picture, if the qubit couples to an environmental operator \(X\) via

then the energy relaxation rate can be written as

where \(S_X(\omega)\) is the symmetrised noise spectral density of \(X\) at the qubit frequency. In practice engineers work to reduce \(S_X(\omega_q)\) by improving materials, reducing loss tangents and engineering high impedance environments.

Purcell decay provides a concrete example. If a qubit is dispersively coupled to a readout resonator of linewidth \(\kappa\) and coupling \(g\), the Purcell limited relaxation time is approximately

where \(\Delta\) is the detuning between qubit and resonator. Simply making \(\Delta\) large or inserting a Purcell filter can lengthen \(T_1\).

Dephasing and \(1/f\) noise

Dephasing in superconducting qubits is dominated by low frequency fluctuations in the qubit frequency, which in turn arise from \(1/f\) flux noise, charge noise and residual photon number fluctuations in the readout resonator [2][4][5]. A widely used model for flux noise assumes a spectral density

with amplitude \(A_\Phi\) of order a few micro flux quanta per root hertz at one hertz [2].

If the qubit frequency depends on flux as \(\omega_q(\Phi)\), then small fluctuations \(\delta \Phi(t)\) translate into frequency noise \(\delta \omega_q(t) = (\partial \omega_q/\partial \Phi) \delta \Phi(t)\). For Gaussian, stationary noise, the free induction decay of a Ramsey experiment can be written as

where \(S_{\delta \omega}(\omega) = (\partial \omega_q/\partial \Phi)^2 S_\Phi(\omega)\). For pure \(1/f\) noise this integral produces approximately Gaussian decay \(\exp[-(t/T_\phi)^2]\) with a dephasing time that scales inversely with \(\abs{\partial \omega_q/\partial \Phi}\). This is why flux and transmon qubits are often operated at "sweet spots" where the first derivatives with respect to flux or charge vanish [1].

Another important dephasing channel is photon shot noise in the readout resonator. Fluctuations in residual photon number shift the qubit frequency via the dispersive interaction, adding phase noise. If the qubit resonator dispersive shift is \(\chi\) and the photon number fluctuations have variance \(\langle \delta n^2 \rangle\), the induced dephasing rate scales roughly as

where \(S_n(0)\) is the low frequency photon number noise spectral density [5].

Trapped ion qubits

In trapped ion platforms, qubits are encoded either in hyperfine levels of the ion ground state or in optical clock transitions. I find it helpful to distinguish internal state decoherence from motional decoherence, because many gates entangle internal states with collective motional modes of the ion crystal.

Motional heating from electric field noise

Ions are confined in radio frequency Paul traps or Penning traps, and their motional modes couple to fluctuating electric fields at the trap electrodes. An empirical observation is that the electric field noise spectral density near electrode surfaces scales approximately as

where \(d\) is the ion electrode distance and \(\alpha\) is typically between zero and two [6][7]. This so called anomalous heating leads to a motional heating rate

where \(\omega_m\) is the motional frequency, \(m\) is the ion mass and \(e\) the elementary charge. Heating of the motion during multi qubit gates reduces gate fidelity, because many high fidelity gates assume the motional mode stays close to its ground state.

This noise is believed to arise from fluctuating adsorbates and patch potentials on the electrode surfaces, and it can be reduced by operating at cryogenic temperatures and by careful surface cleaning [6].

Internal state decoherence

Even if motional modes are perfectly controlled, the internal qubit states suffer from magnetic field noise, laser phase noise and spontaneous emission during Raman or optical gates. For hyperfine qubits, magnetic field noise causes fluctuations in the Zeeman splitting. The associated dephasing rate can be written in analogy with the superconducting case as

where \(S_B(0)\) is the low frequency magnetic field noise spectral density. Operating at "clock transitions" where \(\partial \omega_q/\partial B \approx 0\) can dramatically suppress this dephasing channel.

During laser driven gates, the finite excited state admixture introduces spontaneous emission errors. For a simple Raman scheme with detuning \(\Delta\) and Rabi frequency \(\Omega\), the spontaneous emission error probability during a gate of duration \(\tau\) scales like

where \(\Gamma\) is the natural linewidth of the excited state. Large detuning and high power suppress this error at the cost of technical constraints on laser systems.

Neutral atom qubits

Neutral atom platforms encode qubits in ground state hyperfine levels of neutral atoms trapped in optical tweezers or optical lattices, and entangling gates are typically mediated by Rydberg interactions. The noise landscape looks like a hybrid of trapped ion physics and photonic control noise.

Trap induced dephasing and motional effects

Atoms are confined in optical dipole traps whose intensity fluctuations directly modulate the differential light shift of the two qubit states. If the qubit frequency shift is proportional to trap intensity \(I\) with coefficient \(k\), then

and the resulting dephasing can be treated with the same spectral density framework as for superconducting qubits [8][9]. A particularly important feature here is that atoms move in the trap. Because the light intensity varies across the focus, thermal motion makes each atom sample a fluctuating light shift, even if the laser power is perfectly stable. This creates inhomogeneous broadening and motional dephasing.

Magnetic field noise also contributes, but in many cases it can be partially suppressed by choosing field insensitive hyperfine states, similar to clock transitions in trapped ions [8].

Rydberg state decoherence

Rydberg mediated gates involve transient excitation of atoms to very high principal quantum number states. These states are extremely sensitive to stray electric fields, blackbody radiation and laser noise [9][10]. The total dephasing rate during a Rydberg gate has contributions from

• finite Rydberg lifetime, which sets a fundamental limit on gate duration,
• laser phase noise and amplitude noise that perturb the Rabi oscillations,
• Doppler shifts from atomic motion.

As an example, consider a resonant Rabi oscillation between ground and Rydberg states with Rabi frequency \(\Omega\) and detuning noise \(\delta \Delta(t)\). In the rotating frame the effective Hamiltonian reads

Low frequency fluctuations \(\delta \Delta(t)\) lead to inhomogeneous broadening, reducing the contrast of Rabi oscillations. High frequency components near \(\Omega\) can drive transitions that effectively randomise the phase. Experimental work has shown that carefully reducing laser phase noise can significantly improve Rydberg gate fidelities [10].

Spin qubits in semiconductors

Spin qubits use the spin degree of freedom of electrons or holes confined in quantum dots or donors in materials such as silicon, silicon germanium or gallium arsenide. Decoherence here is dominated by magnetic noise from nuclear spins, charge noise coupling through spin orbit interaction, and sometimes spin phonon coupling.

Hyperfine induced dephasing

In many early spin qubits, such as those in gallium arsenide, the electron spin couples strongly to a bath of nuclear spins. The hyperfine interaction produces an effective Overhauser magnetic field \(B_\text{nuc}\) that fluctuates slowly in time. If the qubit Larmor frequency is \(\omega_q = g \mu_B (B_0 + B_\text{nuc})/\hbar\), then fluctuations of \(B_\text{nuc}\) create frequency noise and inhomogeneous dephasing with \(T_2^*\) times often in the nanosecond to microsecond range. Echo techniques and isotopic purification, for example enrichment of silicon in the spin zero isotope \(^{28}\text{Si}\), can extend coherence times by orders of magnitude [11][12].

A simple model takes \(B_\text{nuc}\) as a quasi static random variable with Gaussian distribution of variance \(\sigma_B^2\). The Ramsey signal then decays as

leading to

Charge noise and \(1/f\) frequency noise

In modern silicon spin qubits with highly purified nuclear environments, dephasing is often limited by charge noise that shifts the electrostatic confinement potential and therefore the \(g\) factor or spin spin coupling via spin orbit interaction [12]. The result is once again low frequency \(1/f\) noise in the qubit splitting, with spectral density

As in the superconducting case, this leads to Gaussian dephasing envelopes and a strong dependence of \(T_2^*\) on the device operating point. Designs that minimise \(\partial \omega_q/\partial V\) with respect to gate voltages and that reduce the sensitivity of the \(g\) factor to electric fields can substantially improve coherence [12].

Relaxation \(T_1\) for spin qubits is usually set by spin phonon coupling mediated by spin orbit interaction. Empirically, \(T_1\) times in the second range have been reported in some silicon devices at millikelvin temperatures, which is very encouraging for quantum error correction.

Photonic qubits

In photonic platforms, information is encoded in individual photons, for example in their path, polarisation, time bin or frequency bin. Because photons interact only weakly with their environment, intrinsic decoherence is small. Instead, the dominant noise processes are loss, mode mismatch and detector imperfections.

Photon loss

Photon loss is the central noise channel in almost all photonic quantum information processors. In a simple model, each optical component, such as a beamsplitter, waveguide segment or fibre, is assigned a transmission probability \(\eta\) between zero and one. If a photon passes through \(N\) such components, the overall survival probability is

In integrated photonics it is common to describe propagation loss as an exponential attenuation

where \(L\) is the propagation length and \(\alpha\) is an attenuation coefficient related to the waveguide loss in decibels per centimetre. Loss not only reduces count rates but also changes the effective quantum state, which can be particularly damaging in boson sampling and Gaussian boson sampling experiments.

Coupling loss at interfaces between chips and fibres or between free space beams and waveguides adds additional effective beam splitters that randomly remove photons from the computational space.

Mode mismatch, dephasing and detector noise

Many photonic protocols rely on high visibility interference at beamsplitters. This requires photons to be indistinguishable in all degrees of freedom. Partial distinguishability due to frequency jitter, timing jitter or imperfect spatial mode overlap reduces interference visibility and can be modelled as mixing the ideal interference unitary with an incoherent classical mixture.

Phase noise in interferometric circuits acts like dephasing. If a path encoded qubit \(\alpha \ket{0} + \beta \ket{1}\) accumulates a random phase \(\phi\) on one arm, then averaging over \(\phi\) with some distribution washes out off diagonal coherences. For small phase fluctuations with variance \(\sigma_\phi^2\), the coherence decays like \(\exp[-\sigma_\phi^2/2]\).

Single photon detectors contribute dark counts and finite detection efficiency. A dark count can be modelled as a spurious photon added to the state at the detector, while finite efficiency can be modelled as an additional loss channel preceding an ideal detector. Both effects lower the effective fidelity of quantum operations in measurement based schemes.

Topological qubits based on Majorana modes

Topological qubits seek to store information non locally in topological degrees of freedom that are insensitive to local perturbations. The leading candidate platform uses Majorana zero modes in topological superconductors. In an idealised picture, braiding operations on these modes implement fault tolerant gates protected by an energy gap. In reality, topological protection is imperfect and several specific noise mechanisms become important.

Quasiparticle poisoning

The dominant decoherence mechanism discussed in the Majorana literature is quasiparticle poisoning. The basic idea is that an extra fermionic quasiparticle tunnels into or out of the superconducting island that hosts the Majorana modes, changing the fermion parity and effectively flipping or dephasing the encoded qubit [13][14]. The rate of these events depends on the density of nonequilibrium quasiparticles above the superconducting gap and on the presence of subgap states.

In simple stochastic models, one treats poisoning events as Poisson processes with rate \(\Gamma_p\). The probability that the parity has not changed after time \(t\) is

so that the associated coherence time is of order \(T_p = 1/\Gamma_p\). Microscopic calculations relate \(\Gamma_p\) to parameters such as the superconducting gap, temperature, quasiparticle diffusion constants and coupling to normal metal leads [13].

Finite size effects and nonadiabatic errors

Ideal Majorana zero modes are perfectly localised at the ends of a topological superconducting segment and have exactly zero energy. In finite systems, there is always some overlap between Majoranas, leading to a small energy splitting \(\varepsilon\) of the encoded qubit. This splitting makes the qubit sensitive to perturbations and leads to dephasing on a time scale

Moreover, braiding operations must be performed slowly compared to the inverse gap to remain adiabatic, but not so slowly that accumulated control noise dominates. Nonadiabatic transitions out of the ground state manifold and Landau Zener type excitations at avoided crossings can lead to leakage errors. These processes are usually modelled with small transition probabilities per braid, which must be kept below fault tolerance thresholds [14].

Putting the pieces together

Although the microscopic details vary widely, I find that many noise models across modalities reduce to a few recurring mathematical structures.

First, almost every platform can be described by a qubit whose frequency fluctuates due to slow classical noise. This gives rise to inhomogeneous dephasing that produces Gaussian or stretched exponential decay in Ramsey experiments. The key parameters are the noise spectral density at low frequencies and the sensitivity of the qubit frequency to control parameters.

Second, most relaxation processes can be captured by Fermi Golden rule expressions involving the noise spectral density at the qubit transition frequency. Here the engineering challenge is to design environments that strongly suppress noise at that frequency while still allowing fast, high fidelity control and readout.

Finally, non Markovian and non Gaussian noise, such as telegraph noise from two level fluctuators in materials or sudden quasiparticle poisoning events, require more refined stochastic models. These are often built by combining master equations with classical jump processes.

When I step back, what encourages me is that many of the most damaging noise sources, like \(1/f\) charge and flux noise or surface electric field noise, have already been reduced significantly in the last decade through better materials and device design. At the same time, cross fertilisation between modalities is very active. For example, ideas about dynamical decoupling and clock transitions travel freely between trapped ions, neutral atoms and spin qubits, while superconducting and Majorana communities both think deeply about superconducting materials and quasiparticle control. I expect this convergence of ideas to be crucial as we move from noisy intermediate scale devices to genuinely fault tolerant quantum processors.

References

1. P. Krantz, M. Kjaergaard, F. Yan, T. P. Orlando, S. Gustavsson and W. D. Oliver, "A quantum engineer's guide to superconducting qubits," Applied Physics Reviews 6, 021318 (2019).
2. S. Sendelbach, D. Hover, A. Kittel, M. Mueck, J. M. Martinis and R. McDermott, "Magnetism in SQUIDs at millikelvin temperatures," Physical Review Letters 100, 227006 (2008).
3. F. Yan et al., "The flux qubit revisited to enhance coherence and reproducibility," Nature Communications 7, 12964 (2016).
4. J. Bylander et al., "Noise spectroscopy through dynamical decoupling with a superconducting flux qubit," Nature Physics 7, 565–570 (2011).
5. F. Yan et al., "Spectroscopy of low frequency noise and its temperature dependence in a superconducting qubit," Physical Review B 85, 174521 (2012).
6. J. Chiaverini and J. Sage, "Trapped ion quantum information processing: Progress and challenges," Journal of Modern Optics 65, 553–576 (2018).
7. P. Schindler et al., "Polarization of electric field noise near metallic surfaces," Physical Review A 92, 013414 (2015).
8. M. Saffman, T. G. Walker and K. Mølmer, "Quantum information with Rydberg atoms," Reviews of Modern Physics 82, 2313–2363 (2010).
9. L. Henriet et al., "Quantum computing with neutral atoms," Quantum 4, 327 (2020).
10. H. Levine et al., "High fidelity control and entanglement of Rydberg atom qubits," Physical Review Letters 121, 123603 (2018).
11. M. Veldhorst et al., "An addressable quantum dot qubit with fault tolerant control fidelity," Nature Nanotechnology 9, 981–985 (2014).
12. J. Yoneda et al., "A quantum dot spin qubit with coherence limited by charge noise and controllable g factor," Nature Nanotechnology 13, 102–106 (2018).
13. T. Karzig, W. S. Cole and D. I. Pikulin, "Quasiparticle poisoning of Majorana qubits," Physical Review Letters 126, 057702 (2021).
14. D. Litinski, "A game of surface codes: Large scale quantum computing with lattice surgery," Quantum 3, 128 (2019), and references therein for noise models of Majorana based qubits.