GHZ state in superconducting quantum circuits
Understanding the Nishimori transition and the GHZ state in superconducting quantum circuits. The ibm_sherbrooke, a 127-qubit IBM Quantum Eagle processor, was used during the simulation tests. In order to generate the GHZ state using their measurement-based protocol, must first separate the processor’s qubits into two groups: the auxiliary qubits, which are used to prepare and measure the GHZ state, and the system qubits, which will comprise the GHZ state. You can see how nous partitioned the processor’s qubits for an extensive simulation, which was conducted on 125 of the 127 qubits in the IBM Sherbrooke, in the graphic below:
As you can see, the auxiliary qubits are coloured grey, while the system qubits that will represent the GHZ state are coloured black. To ensure that no two system qubits are next to each other, grey auxiliary qubits are positioned between each of the black system qubits. The physical links, or “bonds,” that bind each qubit to its closest neighbours are also colored-coded. The bonds are separated into three categories: grey, red, and blue.
For the circuit protocol to be executed, this detail will be crucial. All qubits are initialized into the ground state at the start of the procedure. In an ideal, error-free scenario, one could generate a GHZ state with no mid-circuit measurements or supplementary qubits. To determine the desired long-range ordered state, one can would only initialize the system qubits, entangle each pair of nearby qubits using solely unitary, two-qubit entangling gates, and then measure the qubits at the conclusion of this straightforward circuit. People do not live in a perfect world, of course.
Inevitably, measurement errors, gate faults, and other types of errors cause the system to become out of long-range order or make it impossible to tell if long-range order has been reached. The supplementary qubits come into play here. Let’s examine a tiny subset of the processor’s qubits two system qubits joined by an auxiliary qubit to have a better understanding of their function:
In order to prepare all three qubits in their respective distinct superposition states, let start the protocol by applying distinct Hadamard gates to each qubit separately. By using two qubit gates, it is able to entangle the two system qubits, which is the goal. The auxiliary qubit that stands in between the system qubits separates them, and in superconducting quantum processors, users can only apply two qubit gates to qubits that are directly adjacent to one another. It will need to carry out a basic quantum teleportation experiment in order to join the two system qubits.
First, there are local unitary two-qubit gates to entangle each of the system qubits to the auxiliary qubit. Everyone just require the two system qubits to be entangled at this stage, even though all three qubits are extremely entangled. In order to construct a maximally entangled Bell pair between the two adjacent system qubits, next apply another Hadamard gate to the auxiliary qubit.
By applying the identical entangling procedures to all system qubits and the auxiliary qubits that sit between them, they can expand this tiny, three-qubit example to all system qubits. In particular, can simultaneously apply the two-qubit gates to all qubits via gray-colored bonds first, then to all qubits via red-colored bonds, and lastly to all qubits via blue-colored bonds. By using this technique, one can can keep the circuit’s depth constant and prevent needless complexity from being added. Additionally, all system qubits experience a macroscopic GHZ state, with the auxiliary qubits acting as entanglement bridges.
Two significant questions are brought up by this:
- In reality, how can things confirm that things have moved from chaos to long-range order?
- How can one study this phase transition’s characteristics and ascertain whether it adheres to the well-known Nishimori line?
Simply protectively measuring each auxiliary qubit and feeding the measurement results into a function on a classical computer that decodes the state of the remaining system qubits is all that is required to answer the first question. It confirm that the system qubits have attained the long-range order of the GHZ state with the aid of this classical decoder.
Everyone can manually introduce both coherent errors that skew qubit interactions and incoherent errors that taint the communications channel between the quantum computer and the classical decoder in order to investigate the nature of this phase transition from disordered to ordered.
This allows us to track the evolution of the quantum state when the disorder probability is adjusted. After introducing both kinds of mistakes, the macroscopic behavior of every pair-wise correlation in the system shows that the quantum state travels along the same route as the Nishimori line:
Given that this greatest experiment simulates up to 54 spins across 125 qubits, this study not only illustrates that the phase transition between order and long-range order can be created and manipulated on a quantum computer, but it also shows how one can do it at scale. Using a larger device such as the IBM Quantum Condor, recent theoretical work suggests that this protocol might even grow up to 1,121 qubits.
