Systems and methods for passive quantum error correction

1. An error-transparent two-qubit quantum circuit, comprising:

a first logical qubit formed from a first plurality of physical qubits;

a second logical qubit formed from a second plurality of physical qubits; and

one or more tunable couplers that implement an error-transparent quantum gate that entangles the first plurality of physical qubits with the second plurality of physical qubits, the error-transparent quantum gate being one of a CZZ gate and a XCX gate that operates independently of single errors in the first and second logical qubits.

2. The error-transparent two-qubit quantum circuit of claim 1 ,

the first plurality of physical qubits comprising first and second physical qubits; and

the second plurality of physical qubits comprising third and fourth physical qubits.

3. The error-transparent two-qubit quantum circuit of claim 2 , each of the first and second logical qubits being a very small logical qubit (VSLQ).

4. The error-transparent two-qubit quantum circuit of claim 3 , each of the first, second, third, and fourth physical qubits being a superconducting qubit.

5. The error-transparent two-qubit quantum circuit of claim 4 , wherein:

each of the first, second, third, and fourth superconducting qubits is a transmon;

the first logical qubit further includes a first SQUID for coupling the first and second transmons; and

the second logical qubit further includes a second SQUID for coupling the third and fourth transmons.

6. The error-transparent two-qubit quantum circuit of claim 5 , wherein:

the first and second transmons share a first common bridged ground; and

the third and fourth transmons share a second common bridged ground.

7. The error-transparent two-qubit quantum circuit of claim 3 , each of the first and second tunable couplers being a gmon coupler.

8. The error-transparent two-qubit quantum circuit of claim 3 , further comprising:

first, second, third, and fourth shadow qubits;

a third tunable coupler for coupling the first shadow qubit and the first logical qubit;

a fourth tunable coupler for coupling the second shadow qubit and the first logical qubit;

a fifth tunable coupler for coupling the third shadow qubit and the second logical qubit; and

a sixth tunable coupler for coupling the fourth shadow qubit and the second logical qubit.

9. The error-transparent two-qubit quantum circuit of claim 8 ,

each of the first, second, third, and fourth shadow qubits being a superconducting qubit; and

each of the first, second, third, and fourth physical qubits being a superconducting qubit.

10. The error-transparent two-qubit quantum circuit of claim 9 ,

each of the first, second, third, and fourth shadow qubits being a resonator; and

each of the first, second, third, and fourth physical qubits being a superconducting qubit.

11. A method for implementing an error-transparent quantum gate with first and second logical qubits, the first logical qubit including a first plurality of physical qubits, the second logical qubit including a second plurality of physical qubits, comprising:

off-resonantly driving one or more tunable couplers with one or more corresponding drive signals to entangle the first plurality of physical qubits with the second plurality of physical qubits, the drive signals being configured to implement with the first and second logical qubits an error-transparent quantum gate that operates independently of single errors in the first and second logical qubits, the error-transparent quantum gate being one of a CZZ gate and a XCX gate.

12. The method of claim 11 , further comprising selecting one or more of a duration, frequency, and amplitude of each of the drive signals to implement the error-transparent quantum gate with the first and second logical qubits.

13. The method of claim 11 ,

each of the drive signals being a single-frequency drive; and

further comprising selecting frequencies of the drive signals so that the drive signals are detuned from two-photon resonances in the physical qubits.

14. The method of claim 11 , each of the drive signals comprising a plurality of single-frequency drives.

15. The method of claim 11 , wherein driving the one of more tunable couplers with the corresponding drive signals includes driving the one or more tunable couplers with corresponding oscillating fluxes to generate mutual inductance in each of said one or more tunable couplers.

16. The method of claim 11 , further comprising:

coupling, prior to driving, the first plurality of physical qubits to form therewith a first logical state manifold of the first logical qubit; and

coupling, prior to driving, the second plurality of physical qubits to form therewith a second logical state manifold of the second logical qubit.

17. A method for implementing an error-transparent quantum gate with a logical qubit, the logical qubit including at least first and second physical qubits, comprising:

driving a tunable coupler with a first drive signal to couple the first and second physical qubits; and

driving a degree of freedom of one of the first and second physical qubits with a second drive signal;

wherein the first and second drive signals are configured to apply to the logical qubit an error-transparent quantum gate that operates independently of single-photon errors in the logical qubit, the error-transparent quantum gate being one of a CZZ gate and a XCX gate.

18. The method of claim 17 ,

each of the first and second physical qubits being a transmon;

the degree of freedom being one of a charge degree of freedom and a flux degree of freedom; and

driving the tunable coupler with the first drive signal includes driving a tunable flux-driven coupler with a single-frequency oscillating flux that generates mutual inductance.