Qubit Frequency Allocation

Fixed-frequency processors must assign each qubit a frequency that enables its gates while avoiding collisions with neighbors and defects.

The Frequency-Crowding Problem

In a fixed-frequency architecture, each qubit's frequency is set at fabrication and cannot be tuned. Gates such as cross-resonance require specific frequency relationships between neighbors. Meanwhile, if two coupled qubits, or a qubit and a spectator, land too close in frequency, or if a difference or sum of frequencies hits a spurious transition, gates fail and idle qubits suffer crosstalk. As qubits are packed together, the usable frequency band fills up and collisions become likely. This is frequency crowding.

Collision Conditions

• Two coupled qubits at the same frequency exchange excitations uncontrollably.
• A qubit frequency equal to a neighbor's second-transition frequency drives leakage.
• Cross-resonance requires the control to sit above the target by roughly the anharmonicity, a window that must not be missed.
• Chains of three qubits can have accidental resonances across the pair.

Allocation as Optimization

Choosing frequencies for a whole lattice is a constrained optimization: assign each qubit a target frequency so that every gate's frequency condition is satisfied and no collision condition is met, given the connectivity graph. The problem resembles graph coloring, and for dense two-dimensional lattices it can be tight or even infeasible within the available band, which is one reason many designs turn to tunable elements instead.

Coping With Fabrication Spread

Even a good allocation is undermined if fabrication cannot hit the target frequencies. Junction variability scatters real frequencies away from their design values, so some fraction of gates fall into collision. Remedies include post-fabrication trimming, such as laser annealing to move junctions toward target, statistical yield modeling to predict how many devices will be collision-free, and choosing lattice connectivity and gate schemes that are more tolerant of spread.

Why Tunability Is an Alternative

Frequency crowding is the price fixed-frequency devices pay for their excellent coherence. Tunable qubits and tunable couplers sidestep it by moving frequencies in situ, at the cost of added flux hardware and flux-noise dephasing. The choice between careful frequency allocation and in-situ tunability is one of the defining forks in superconducting architecture.