Dynamic Circuit Constructions on Braket

Dynamic Circuit Constructions

In this notebook we look at examples of dynamic circuit primitives, including qubit teleportation protocols, ancillary assisted entanglement, remote CNOT applications, which can be run on IQM Garnet with Amazon Braket.

Notebook Setup

Before we dive into our examples, let's handle our imports. For simplicity, we will work here with standard Clifford gates, Pauli rotations, and cc_x or cc_z primitives. The cc_x gate (and by a simple rotation cc_z) applies a X gate conditionally on a particular measurement configuration.

In the local simulator, we use a composition of a partial trace and initialization procedure, which are both valid quantum channels and thus can be implemented through their Kraus operators ():

Notes on iqm_config.py and local_config.py:

• cc_x and cc_z are user defined gate types. Use the reset keyword to specify an active reset.
• These gates assume that measure_ff and any cc_prx are implemented at the same layer and time, and can be concatenated as cc_x.
• Importing iqm_config.py transpiles Clifford circuits to native IQM gates.
• For actual implementations, see the respective iqm_config.py or local_config.py files.

Teleporting Qubits

By utilizing the principle of deferred measurement with qubit swapping techniques, we can derive the following relations:

which is referred to as Z-teleportation. Using Z-teleportation, we can shuffle a qubit through a sequence of swaps, at a potentially reduced cost over unitary swap sequences. This is implicitly used in some long-range CNOT sequences, and represents a fundamental tool in the dynamical circuit toolkit. Another fundamental tool is X-teleportation:

which is derived by teleporting with Z-teleportation and then applying after the teleportation.

In the example here, we use X-teleportation along a list of qubits, and then show that the final target qubit state matches the input qubit state, ideally with coherences preserved. Here we will perform single-qubit tomography on the standard -basis populations of each qubit, which should only be 1 for the last qubit.

• Note, to recover the 0 state in the ancilla, an active reset must be used. If this is not performed, we only teleport the target state, we do not preserve the 0 ancilla state.

Despite noise on the device, we can see the state being transferred succesffuly throughout the protocol. Additionally, because we do not perform active qubit resets, the discarded qubits yield an equal statistical mixture after being used.

Extending CNOT Connectivity with Ancilla Qubits

Here, we look at creating entanglement using dynamical circuits for multi-qubit operations, particularly with a variant of the PARITY gate. Consider the layout on IQM Garnet:

Taking qubits 2,4,6, and 10, which generally could be connected to others qubits, we can attempt to apply a limited double excitation primitive (related to a PARITY gate, but simply a Pauli rotation). For four qubits there are generally straightforward ways of doing things, but disconnected qubits and limited connectivity can complicate this. Furthermore, for PARITY gates and related circuit primitives, there are alternate decompositions which can be implemented using dynamical circuits at constant depth or limited width.

Here, we can realize the Pauli operation by swapping the parity qubit to a joint ancilla (qubit 5) which connects all the qubits, and then swapping it back.

The dominant states here are the and states, which are rotated under a Pauli gate.

Gate Teleportation for Alice and Bob

Consider Alice and Bob, each with their own qubit states. If we want to apply a remote CNOT between their states, without accessing their state directly, we can do so by sharing an intermediate entangled qubit. While there might be some limitations to the specific type which can be applied, there is a realization which exists compatible with the IQM features. Note, this follows C.2. in Appendix of this arXiv preprint.

We first prepare Alice and Bob's states. Then, Alice swaps her state with an ancilla, and uses X-teleportation to send the ancilla to Bob. Bob applies a CNOT and then teleports the circuit back using Z-teleportation. Because not all of these can be accomplished using local CNOTs, one can commute the expressions in such a way such that only a shared entangled resource is required, and the rest can be handled with dynamic circuits.

Predicted Costs

To run these locally costs....nothing! However, you can try these out on IQM Garnet for the expected value listed in the next block.

Conclusions

In this notebook we explored basic circuit structures for dynamic circuit constructions using IQM Garnet on Amazon Braket. These represent fundamental tools for dynamic circuit generation, and allow for more complex and detailed applications.

References:

1. Zhou, Leung, Chaun. Methodology for quantum logic construction (2000) arXiv:0002039.
2. Buhrman, Folkertsma, Loff, Nuemann. State preparation by shallow circuits using feed forward (2024) Quantum 8, 1552.
3. Bäumer, Tripathi, Wang, et al. Efficient Long-Range Entanglement Using Dynamic Circuits (2024) PRX Quantum, 5 030339.