Replacement-Type Quantum Gates

TL;DR

This work introduces replacement-type gates, a paradigm that replaces qubits at the input with output qubits constructed from auxiliary candidate qubits via a rearrangement in an extended Hilbert space that includes particle positions. By avoiding Bloch-sphere rotations and leveraging position degrees of freedom, these gates can approximately preserve hardware noise bias, making them attractive for biased-noise quantum error correction schemes. The authors develop concrete protocols for spin-qubit quantum dots and for neutral-atom qubits, detailing the required primitives (PSB, energy-selective tunneling, state-dependent trapping, Rydberg interactions, and occupation-controlled tunneling) and providing error analyses and performance estimates through simulations and toy fidelities. Their results suggest that replacement-type gates can operate near bias-preserving fault-tolerance thresholds, enabling potentially lower QEC overhead, especially when integrated with biased codes like the XZZX surface code; they also discuss practical aspects, such as gate durations, fidelity targets, and erasure-based mitigation strategies for leakage. This framework opens a route to platform-flexible, bias-respecting quantum gates with implications for architecture design and fault-tolerant quantum computing.

Abstract

We introduce the paradigm of replacement-type quantum gates. This type of gate introduces input qubits, candidate qubits, and output qubits. The candidate qubits are prepared such, that a displacement conditional on the input qubit results in the targeted output state. Finally, the circuit continues with the output qubits constructed from the candidate qubits instead of the input qubits, thus the name "replacement-type gate". We present examples of replacement-type $X$ and $\mathrm{CNOT}$ gates realized with spin qubits and with neutral atom qubits with error rates predicted near the threshold of the XZZX surface code. By making use of the extended Hilbert space, including the position of the particles, these gates approximately preserve the innate noise bias of the qubits. The gate preserves the noise bias which motivates advanced quantum computer architectures with quantum error correction.

Figures (10)

• Figure 1: Concept of the replacement-type single-qubit gate $U$. An input qubit has the general qubit state $|q \rangle = \alpha_0 |0 \rangle + \alpha_1 |1 \rangle$ and is placed at the yellow input site. Two candidate qubits are initialized in suitable basis states of the Hilbert space, here in the states $|0_U (1_U) \rangle = U|0(1) \rangle$, and are placed at the blue auxiliary sites, the green output site is initially empty, $|x\rangle$. Then, the register is rearranged coherently in order to transfer one of the candidate qubits into the output site, conditional on the state of the input qubit. This leads to an entangled state of all four qubits and the configuration of the positions. The amplitude $\alpha_0$ ($\alpha_1$) is now associated with state $|0_U\rangle$ ($|1_U \rangle$) at the output site, imposing the effect of the gate U. Disentangling the population of the output site from the rest of the register with suitable measurements leaves a single qubit in the final separable state $|\psi_{0,1,2}^{(d)}\rangle \otimes U|q \rangle$, corresponding to a gate $U$ and a relocation of the qubit state.
• Figure 2: (a) Sketch of two QDs in a Si/SiGe heterostructure. The electrical potential applied to plunger (P) and barrier (B) gates defines potential wells, which can be filled with single electrons or holes by coupling them to source (S) and drain (D) reservoirs. A strong magnetic field inhomogeneity, also referred to as synthetic spin-orbit interaction, is commonly created by a micromagnet. (b) Energy levels of two QDs, left ($0$) and right ($1$) occupied by a single spin. In each QD, two spin states $|\!\downarrow\rangle$ and $|\!\uparrow\rangle$ are available, with a splitting corresponding to the total magnetic field $B_{0(1)}$ (red). A transverse magnetic gradient $B_{0,x} - B_{1,x} \neq 0$ leads to non-collinear quantization axes. The energy detuning between the ground states, $E_0-E_1$ (blue), is set by the electrical potentials applied to the plunger gates. The voltage applied to the barrier gates tunes the tunneling between the QDs (teal) with a spin-conserving ($t_c$) and spin-flipping ($t_f$) matrix element.
• Figure 3: Spectrum of a double QD in an inhomogeneous magnetic field occupied with no (purple), one (blue) or two (red) electrons as a function of the energy detuning. We label the states as $(\sigma_0 , \sigma_1)$ with the spin projection $\sigma_{0(1)}$ in QD $0$($1$), $x$ indicates an empty QD. With a single electron, avoided level crossings (ALCs) are opened at zero detuning by the tunneling matrix elements, far from the ALCs spin and position of the electron are good quantum numbers. The asymmetry is due to the longitudinal magnetic gradient. Adding a second particle leads to an offset in energy due to Coulomb repulsion, the two electrons can occupy a single QD only if their spins form a singlet ($S$). Due to the longitudinal magnetic field gradient the singlet and triplet states decompose into the product states in the regime with one electron per QD.
• Figure 4: Protocol for a replacement-type $X$ gate with spin qubits. The initial configuration of the array of four QDs is depicted in the top, $q$ indicates the input qubit, $S$ the two-qubit singlet state. Spin-dependent (red arrows) and energy-selective (teal arrows) tunneling are used for manipulating the register, lines 1-4 show the configuration after each step of the protocol. The two columns correspond to the classically distinguishable qubit states $q= \uparrow$ (right) and $q=\downarrow$ (left), in general, a superposition of the columns is obtained. A crossed-out arrow indicates suppressed tunneling. The final output qubit is found at the output site QD 2, highlighted in red. Step 4 is explained in more detail in Sec. \ref{['sec_spins_disentangle']} and Fig. \ref{['fig_spins_disentangle']}.
• Figure 5: Example of a disentangling operation for spin qubits following an $X$ gate. The top row shows the final configuration of the QD array after the rearrangement from Fig. \ref{['fig_spins_X']} for both input states, with the output qubit from QD 2 shuttled away. Steps 1 and 2 prepare a configuration where a single measurement is sufficient for the disentangling, the figure shows the configuration after each step. In step 3, the charge of QDs 1 and 2 is measured in the basis defined by $\left( |1_1\rangle |x_2\rangle \pm |x_1\rangle |1_2\rangle \right)/\sqrt{2}$, where $1(x)$ indicates a QD occupied by one (zero) electrons. A measurement of the total charge occupation in QD 1 or 2 would be equivalent to measuring the output qubit.