Exponential-SWAP (eSWAP) Gate

• Exponential-SWAP (eSWAP) is a continuously tunable two-qubit unitary defined as exp(–iθ SWAP) that smoothly interpolates between the identity and full SWAP operation.
• It is implemented either directly using a flux pulse on a tunable coupler or via an ancilla-based protocol, achieving gate times around 20–30 ns with fidelities exceeding 98%.
• The method generalizes to multi-qubit exchange networks and even to numerical quadrature schemes for near-singular integral evaluations, highlighting its algorithmic and practical versatility.

Exponential-SWAP (eSWAP) denotes, in one usage, the continuous two-qubit unitary family $$U_{\mathrm{eSWAP}}(\theta)=e^{-i\theta\,\mathrm{SWAP}}$$, which interpolates between the identity and the SWAP gate through a continuously tunable exchange angle. In the cited literature, the same label is also used for an exponential-basis extension of singularity-swap quadrature for nearly singular line integrals on closed curves in two dimensions. The shared acronym is therefore best read contextually: in quantum information it refers to exponentiation of the SWAP operator, whereas in numerical analysis it refers to swapping a target singularity to a point close to the unit circle and evaluating the resulting integral in a complex-exponential basis (Rasmussen et al., 2020, Klinteberg, 2023).

1. Two-qubit definition and algebraic action

For two qubits, the SWAP operator is the unitary

$\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$

in the computational basis $\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}$. As a Hermitian involution $\mathrm{SWAP}^2=I$, it has two eigenvalues $+1$ and one eigenvalue $-1$. By spectral calculus one defines the “exponential-SWAP” family

$$U_{\mathrm{eSWAP}}(\theta)\coloneqq e^{-i\theta\,\mathrm{SWAP}} = \cos\theta\,I - i\sin\theta\,\mathrm{SWAP}.$$

Its action is diagonal on $|00\rangle$ and $|11\rangle$,

$$U_{\mathrm{eSWAP}}(\theta)|00\rangle = e^{-i\theta}|00\rangle,\qquad U_{\mathrm{eSWAP}}(\theta)|11\rangle = e^{-i\theta}|11\rangle,$$

while on the single-excitation subspace it acts as

$\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$0

$\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$1

Up to the overall phase $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$2, this is exactly a rotation in the $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$3 subspace by angle $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$4, passing continuously from the identity $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$5 to the SWAP gate $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$6 (Rasmussen et al., 2020).

An equivalent description uses a natural two-qubit exchange Hamiltonian. In systems with an XY-type interaction

$\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$7

the time evolution

$\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$8

coincides, up to local $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$9-phases, with a fractional $\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}$0SWAP on the single-excitation subspace. Setting $\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}$1 realizes the continuously tunable exchange block that underlies the eSWAP construction.

2. Deterministic realization via XY exchange

A direct implementation uses two transmons connected by a tunable coupler. One applies a flux pulse on the coupler that brings the two target qubits into resonance for a duration

$\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}$2

where $\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}$3 is the effective exchange-coupling strength. In a frame that removes the single-qubit energies and absorbs any dynamic $\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}$4-shifts into later calibrated $\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}$5-rotations on each qubit, the time evolution is

$\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}$6

so the native interaction is a fractional $\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}$7SWAP (Rasmussen et al., 2020).

To convert this fractional $\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}$8SWAP into exactly $\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}$9, one inserts local $\mathrm{SWAP}^2=I$0-rotations $\mathrm{SWAP}^2=I$1 on each qubit before and after the exchange block. In practice one calibrates the net single-qubit phases and performs them in software as virtual $\mathrm{SWAP}^2=I$2 gates. The resulting implementation is deterministic, has depth $\mathrm{SWAP}^2=I$3, and uses a single flux-pulse window together with two virtual $\mathrm{SWAP}^2=I$4’s per side.

The same description gives a concrete duration scale: for $\mathrm{SWAP}^2=I$5 MHz one finds $\mathrm{SWAP}^2=I$6, so a $\mathrm{SWAP}^2=I$7 at $\mathrm{SWAP}^2=I$8 is $\mathrm{SWAP}^2=I$9 ns. The paper’s central hardware point is that a simple single-flux-pulse on a tunable coupler turns the coupling on for time $+1$0 and directly implements the fractional $+1$1SWAP, which by local $+1$2-rotations can be converted into the eSWAP above.

3. Ancilla-based exponentiation and branch correction

As an alternative, the cited work extends Marvian & Lloyd’s method to cyclic non-Hermitian gates. Since $+1$3, SWAP is a Hermitian involution of cyclic order $+1$4. The protocol introduces one ancilla qubit, prepares it in the superposition

$+1$5

applies one controlled-SWAP (Fredkin) gate controlled by the ancilla and targeting the two data qubits, and then measures the ancilla in the $+1$6-basis $+1$7, with

$+1$8

If the outcome is $+1$9, the two-qubit register is projected onto

$-1$0

so one exactly realizes $-1$1 up to a known global normalization (Rasmussen et al., 2020).

If instead the result is $-1$2, one obtains

$-1$3

which can be converted back to the target $-1$4 by a simple corrective routine, either a second controlled-SWAP or a post-measurement phase flip. The feed-forward for the “$-1$5” branch is described as trivial: because the outcome differs by an extra minus-sign on the SWAP term, a single controlled-$-1$6 or a second controlled-SWAP yields the desired sign flip.

The success probability for the “$-1$7” branch is

$-1$8

For small $-1$9 this is $$U_{\mathrm{eSWAP}}(\theta)\coloneqq e^{-i\theta\,\mathrm{SWAP}} = \cos\theta\,I - i\sin\theta\,\mathrm{SWAP}.$$0, and at $$U_{\mathrm{eSWAP}}(\theta)\coloneqq e^{-i\theta\,\mathrm{SWAP}} = \cos\theta\,I - i\sin\theta\,\mathrm{SWAP}.$$1 one has $$U_{\mathrm{eSWAP}}(\theta)\coloneqq e^{-i\theta\,\mathrm{SWAP}} = \cos\theta\,I - i\sin\theta\,\mathrm{SWAP}.$$2. With one additional Fredkin, or a controlled-$$U_{\mathrm{eSWAP}}(\theta)\coloneqq e^{-i\theta\,\mathrm{SWAP}} = \cos\theta\,I - i\sin\theta\,\mathrm{SWAP}.$$3 plus local SWAP, the protocol can be made fully deterministic.

4. Resource counts, fidelity, and simulated performance

The direct and ancilla-based realizations differ in qubit overhead and control primitives, but both were analyzed within the same superconducting-circuit setting. The direct implementation uses $$U_{\mathrm{eSWAP}}(\theta)\coloneqq e^{-i\theta\,\mathrm{SWAP}} = \cos\theta\,I - i\sin\theta\,\mathrm{SWAP}.$$4 data qubits, $$U_{\mathrm{eSWAP}}(\theta)\coloneqq e^{-i\theta\,\mathrm{SWAP}} = \cos\theta\,I - i\sin\theta\,\mathrm{SWAP}.$$5 flux-pulse window, two virtual $$U_{\mathrm{eSWAP}}(\theta)\coloneqq e^{-i\theta\,\mathrm{SWAP}} = \cos\theta\,I - i\sin\theta\,\mathrm{SWAP}.$$6’s per side, depth $$U_{\mathrm{eSWAP}}(\theta)\coloneqq e^{-i\theta\,\mathrm{SWAP}} = \cos\theta\,I - i\sin\theta\,\mathrm{SWAP}.$$7, and duration $$U_{\mathrm{eSWAP}}(\theta)\coloneqq e^{-i\theta\,\mathrm{SWAP}} = \cos\theta\,I - i\sin\theta\,\mathrm{SWAP}.$$8. The probabilistic implementation uses $$U_{\mathrm{eSWAP}}(\theta)\coloneqq e^{-i\theta\,\mathrm{SWAP}} = \cos\theta\,I - i\sin\theta\,\mathrm{SWAP}.$$9 additional ancilla, one controlled-SWAP (Fredkin) gate, two single-qubit rotations on the ancilla, one ancilla readout in the $|00\rangle$0-basis, and post-selection or one correction Fredkin for determinism (Rasmussen et al., 2020).

Method	Resources	Performance statement
Direct via coupler pulses	2 data qubits; 1 flux-pulse window; virtual $|00\rangle$1 gates	Gate times are $|00\rangle$2–$|00\rangle$3 ns
Ancilla-based	1 ancilla; 1 Fredkin; ancilla rotations and readout	Overall exponentiation fidelities $|00\rangle$4 for small $|00\rangle$5

The fidelity analysis in Rasmussen and Zinner includes realistic qubit coherence times $|00\rangle$6s, flux-pulse shaping to avoid leakage, and random $|00\rangle$7–$|00\rangle$8 fabrication scatter in circuit parameters using QuTiP master-equation simulations. For the bare controlled-$|00\rangle$9SWAP $|11\rangle$0 with $|11\rangle$1, the reported process fidelities are above $|11\rangle$2 in the noiseless case, and $|11\rangle$3 once decoherence is included. Even for up to four controls $|11\rangle$4 the average fidelity stays $|11\rangle$5 under the same $|11\rangle$6 assumptions. A large Monte-Carlo ensemble with $|11\rangle$7 jitter in Josephson energies and capacitances still yields $|11\rangle$8 of runs above $|11\rangle$9 final fidelity.

The same hardware can directly implement the fractional eSWAP via the XY exchange pulse with identical performance numbers. Alternatively, the ancilla-based protocol inherits the underlying controlled-SWAP fidelity $$U_{\mathrm{eSWAP}}(\theta)|00\rangle = e^{-i\theta}|00\rangle,\qquad U_{\mathrm{eSWAP}}(\theta)|11\rangle = e^{-i\theta}|11\rangle,$$0 per Fredkin plus single-qubit gates at $$U_{\mathrm{eSWAP}}(\theta)|00\rangle = e^{-i\theta}|00\rangle,\qquad U_{\mathrm{eSWAP}}(\theta)|11\rangle = e^{-i\theta}|11\rangle,$$1 each, so overall exponentiation fidelities $$U_{\mathrm{eSWAP}}(\theta)|00\rangle = e^{-i\theta}|00\rangle,\qquad U_{\mathrm{eSWAP}}(\theta)|11\rangle = e^{-i\theta}|11\rangle,$$2 are readily within reach for small $$U_{\mathrm{eSWAP}}(\theta)|00\rangle = e^{-i\theta}|00\rangle,\qquad U_{\mathrm{eSWAP}}(\theta)|11\rangle = e^{-i\theta}|11\rangle,$$3.

5. Generalizations and algorithmic relevance

The superconducting-coupler toolbox extends beyond a single pair of qubits. Section IV of the cited work shows how to wire up an all-to-all exchange network with each pair of “swapping” qubits gated by its own ancilla or tunable-bus element. One can build $$U_{\mathrm{eSWAP}}(\theta)|00\rangle = e^{-i\theta}|00\rangle,\qquad U_{\mathrm{eSWAP}}(\theta)|11\rangle = e^{-i\theta}|11\rangle,$$4 or eSWAP operations between any chosen pair, or even three-way or four-way swaps, by the same “turn-on–turn-off” flux control (Rasmussen et al., 2020).

The circuit-based exponentiation protocol also generalizes from SWAP to arbitrary cyclic operators. For a general cyclic gate of order $$U_{\mathrm{eSWAP}}(\theta)|00\rangle = e^{-i\theta}|00\rangle,\qquad U_{\mathrm{eSWAP}}(\theta)|11\rangle = e^{-i\theta}|11\rangle,$$5, satisfying $$U_{\mathrm{eSWAP}}(\theta)|00\rangle = e^{-i\theta}|00\rangle,\qquad U_{\mathrm{eSWAP}}(\theta)|11\rangle = e^{-i\theta}|11\rangle,$$6, the paper shows that one needs $$U_{\mathrm{eSWAP}}(\theta)|00\rangle = e^{-i\theta}|00\rangle,\qquad U_{\mathrm{eSWAP}}(\theta)|11\rangle = e^{-i\theta}|11\rangle,$$7 ancillas and $$U_{\mathrm{eSWAP}}(\theta)|00\rangle = e^{-i\theta}|00\rangle,\qquad U_{\mathrm{eSWAP}}(\theta)|11\rangle = e^{-i\theta}|11\rangle,$$8 controlled-$$U_{\mathrm{eSWAP}}(\theta)|00\rangle = e^{-i\theta}|00\rangle,\qquad U_{\mathrm{eSWAP}}(\theta)|11\rangle = e^{-i\theta}|11\rangle,$$9 operations, measured in the $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$00 basis. The success probability of the all-plus branch scales as $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$01 for small $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$02, but may be improved by feed-forward. A specific example in the paper is controlled-$\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$03SWAP itself, which has order $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$04; accordingly, one may build $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$05 exactly with three ancillas and $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$06 controlled-$\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$07SWAPs.

Fractional SWAPs arise in algorithms for quantum walks, Hamiltonian simulation, entanglement distribution and CV-cluster state generation. The cited discussion states that having a native, high-fidelity eSWAP that is continuously tunable in $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$08 can dramatically reduce Trotter-step counts and gate depths in near-term devices. Within that framing, eSWAP is not merely a gate interpolation, but a hardware-level exchange primitive that can be invoked either directly through analog XY coupling or exactly through ancilla-assisted circuit exponentiation.

6. Distinct numerical-analysis usage: exponential-swap quadrature

In a separate numerical-analysis literature, “Exponential-Swap” or eSWAP denotes a quadrature method for evaluating nearly singular line integrals in two dimensions close to periodic boundaries, discretized using the trapezoidal rule. The problem is to evaluate, for a target point $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$09 identified with $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$10 close to a smooth closed curve $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$11, a layer-potential integral

$\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$12

with $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$13 parameterized by a $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$14-periodic analytic map $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$15. As $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$16, the integrand ceases to be smooth because there is a nearby singularity at $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$17 satisfying $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$18, and standard trapezoidal-rule convergence deteriorates like $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$19 (Klinteberg, 2023).

The key idea is threefold: find the complex preimage $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$20 via Newton’s method on the Fourier-interpolant of $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$21; swap the singular factor $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$22 for a factor $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$23, leaving a smooth remainder $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$24; and expand $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$25 in the basis $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$26 and evaluate the remaining singular integral analytically using contour integrals on the unit circle. Because $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$27 is smooth and $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$28-periodic, one samples it at $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$29, computes discrete Fourier coefficients by an FFT, and then approximates

$\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$30

where the weights

$\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$31

are obtained analytically. For $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$32 and $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$33,

$\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$34

for $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$35, and $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$36 for $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$37.

The implementation described in the paper precomputes node values $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$38 and $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$39 and their forward FFT’s, finds $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$40 starting from the real node whose $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$41 is closest to $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$42, performs $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$43–$\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$44 steps of Newton’s method, forms the regularized samples

$\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$45

computes $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$46 by FFT in $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$47, and evaluates the final dot product with the explicit residue-theorem formula for $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$48. The total cost per target is $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$49. To achieve a relative precision $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$50, one needs roughly

$\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$51

The theoretical error estimate is

$\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$52

because $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$53 is analytic in the strip $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$54 and its Fourier coefficients decay like $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$55. Empirical results on the starfish geometry $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$56 with $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$57 up to $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$58 show that standard trapezoidal-rule suffers $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$59 loss in accuracy as $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$60, whereas eSWAP attains $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$61 accuracy uniformly down to distances as small as $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$62, with exponential convergence in $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$63 independently of distance, and takes $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$64 per target at $\mathrm{SWAP}\,|a,b\rangle = |b,a\rangle$65 on a laptop. In this usage, eSWAP is therefore a singular quadrature scheme rather than a quantum gate, despite the identical acronym.