Measuring $T_1$ time
Learn how to measure the relaxation time of a qubit.
Objectives
- Understand the concept of $T_1$ time and its significance in quantum computing.
- Experimentally measure the $T_1$ time of a qubit.
Background
Theory
The $T_1$ time of a qubit, often referred to as the energy relaxation time or longitudinal relaxation time, is a crucial parameter in the field of quantum computing. It represents the characteristic time scale over which a qubit will decay from the excited state $|1\rangle$ to the ground state $|0\rangle$, and it is called longitudinal because it describes depolarization along the qubit quantization axis. In principle, also the inverse process could happen (excitation from the ground state to the excited state), but generally this is negligible in superconducting implementations.
The relaxation process is driven by the qubit's interaction with its surrounding environment. Thus, the $T_1$ time is determined by the mechanisms of energy dissipation within the system, which are influenced by various environmental factors such as thermal fluctuations, electromagnetic fields, and material defects.
$T_1$ is often associated to another characteristic time $T_2$, often called dephasing time, which instead represents the loss of coherence, and so the loss of quantum behaviour for the system. The combination of the two phenomena is generally referred to as quantum decoherence, i.e. the loss of information from the system to the environment; this played an important role in fundamentals concepts of quantum mechanics like quantum-to-classical transition or the measurement problem.
In quantum computing, the $T_1$ time sets a practical limit on the duration for which a qubit can reliably store quantum information. The longer the $T_1$ time, the better the qubit is at preserving the integrity of its quantum state against energy relaxation, thereby enhancing the potential for performing more advanced quantum algorithms. This is why the errors caused by relaxation have to be carefully taken into account in any realistic noise model of NISQ devices and why $T_1$ is important for performance benchmarking.
Experimentally, it can be measured by preparing the qubit in its excited state and monitoring the probability of the qubit being in this state as a function of time $t$. More precisely, we measure the qubit after different fixed delays of time, and repeating the preparation, waiting and measurement process several times, we can estimate the probability to measure the excited state. If we assume that there are no preparation and measurement errors (the so-called SPAM errors), then the decay of this probability $p$ is exponential and characterized by $p(t) = e^{-\frac{t}{T_1}}$.
Pulse-level access
The experiment to compute the $T_1$ time for the different qubits in our quantum computer involves access to the hardware through the pulse-level interface. This is often done for calibration tasks where one tries to assess fidelities of quantum gates that can compromise the quality of the results of an experiment. Analogously, control pulses become relevant when we want to manipulate the qubits like in the task of preparing them in an excited state.
In general, any operation acting on a qubit is, down to the physical level, characterized by the time evolution of the quantum system. This evolution can be unitary or not, and it depends on whether the quantum system is well isolated. If it is, we call it a closed system, if not we refer to it as an open system, i.e. one that interacts with an environment.
In the first case, the system undergoes a unitary time evolution governed by an Hamiltonian with a controllable and a non-controllable (so-called drift) component; one can modulate the time-dependent control coefficients to drive the system to realize the desired unitary gate. In this way we obtain a pulse-level description of the quantum circuit we want to realize.
In the real world though it is very hard to perfectly isolate a quantum system and thus the corresponding time evolution becomes non-unitary (the Schroedinger equation does not hold anymore!). One needs to include in the description the possible interactions with the environment, an idea that gave rise the the extensive theory of open quantum systems governed by the Lindblad master equation. Delving into the details of this theory is out of the experiment scope, but one important observation that helps clarifying the roles of $T_1$ and $T_2$ is that in this case we use the density matrix to describe the quantum system. For the evolution of a single qubit, the relaxation times characterize respectively the decay of the diagonal entries (so-called populations) and of the off-diagonal ones (so called coherences).
Assignment
- Prepare a qubit in its excited state and measure the probability of it being in this state as a function of time .
- Plot the probability data and observe the exponential decay behavior. Fit the data to an exponential decay curve and extract the $T_1$ time.
- Repeat the process for other qubits in the system and compare the $T_1$ times.
- Compare the results to the $T_1$ from other qubit implementations (like trapped ions, cold atoms, etc.) from literature. Put it in relation to the gate times of the different architectures.
Questions for Preparation
Why is $T_1$ important for the execution of advanced routines?
What are the factors that could contribute to the differences in $T_1$ times between different qubit implementations?
How would the decay look like in the presence of SPAM errors?
What is the biggest difference between closed and open quantum system dynamics?
How can we represent quantum systems in open dynamics?
Literature
Superconducting qubits, William D. Oliver (Lecture Notes).
Pulse-level noisy quantum circuits with QuTiP (Quantum 6, 630 (2022)).
Modelling and Simulating the Noisy Behaviour of Near-term Quantum Computers (Phys. Rev. A 104, 062432).