Non-Markovian Quantum Process Tomography

Characterization protocols have so far played a central role in the development of noisy intermediate-scale quantum (NISQ) computers capable of impressive quantum feats. This trajectory is expected to continue in building the next generation of devices—ones that can surpass classical computers for particular tasks—but progress in characterization must keep up with the complexities of intricate device noise. A missing piece in the zoo of characterization procedures is tomography, which can completely describe non-Markovian dynamics over a given time frame. Here, we formally introduce a generalization of quantum process tomography, which we call process tensor tomography. We detail the experimental requirements, construct the necessary postprocessing algorithms for maximum-likelihood estimation, outline the best-practice aspects for accurate results, and make the procedure efficient for low-memory processes. The characterization is a pathway to diagnostics and informed control of correlated noise. As an example application of the hardware-agnostic technique, we show how its predictive control can be used to substantially improve multitime circuit fidelities on superconducting quantum devices. Our methods could form the core for carefully developed software that may help hardware consistently pass the fault-tolerant noise threshold.

Popular Summary

Quantum computation promises to change the paradigm of computing and research. It requires absolute control over a lattice of quantum bits, or qubits, through logical operations called gates. To gain an advantage over current classical computing, it is critical that these gates have as little error in them as possible. Unfortunately, qubits are always coupled to their environment and errors can occur probabilistically.

Sometimes this is simple. But imagine that you were to roll a die several times in a row by placing it on a surface and shaking. However, instead of equally weighted outcomes, some external agent came and shook the surface so that now the probability depended on whether you last rolled a 1, then 5, and then a 3. These “memory” effects are called non-Markovian and they make things a lot more complicated. Similarly, in quantum computing, the behavior of noise may depend on the choice of the last several gates. This can become extremely complex and is not very well understood.

We provide a framework and a recipe for characterizing these non-Markovian dynamics. What this allows an experimenter to do is things such as benchmarking and asking “How non-Markovian is my device?” but, also, the characterization says exactly what the device will do in response to any sequence of gates. We show by way of demonstration that this permits much cleaner computation with IBM Quantum superconducting devices. Widespread implementation of this characterization and control could see significant improvement in the abilities of current quantum devices.

Figure 1

(a) The conventional model of open quantum systems tracks the state of the system as a function of time, but cannot build up multitime joint statistics. (b) The quantum stochastic process picture considers the response of the system to different sequences of gates; by considering all trajectories of the system, correlations between different times may be exactly characterized and quantified. (c) A pictorial description of how linear expansion in a basis by the process tensor preserves intermediate dynamics and expresses an arbitrary sequence.

Figure 2

Operation, manipulation, and characterization analogs between quantum process tomography and process tensor tomography. (a) The Choi-Jamiolkowski isomorphism represents a quantum process $\mathcal{E}$ by the density matrix $\hat{\mathcal{E}}$ using the action of $\mathcal{E}$ on one half of a maximally entangled state. (b) In the generalized CJI, $SE$ unitaries act on one half of a maximally entangled state per time step. Correlations between times (denoted using red dashed lines) are then mapped onto spatial correlations between each output leg of a Bell pair. The result is a collection of (possibly correlated) CPTP maps, as well as the average initial state. (c) The outcome of $\mathcal{E}$ on some ${\rho }_{\mathrm{in}}$ is obtained by projecting the Choi state onto $\mathbb{I}\otimes {\rho }_{\mathrm{in}}^T$ and tracing over the input space. (d) The outcome of a process conditioned on a sequence of operations ${\mathbf{A}}_{k-1:0}$ is obtained by projecting the Choi state of ${\mathrm{{\rm Y}}}_{k:0}$ onto the Choi state ${⨂}_{i=0}^{k-1}{\mathcal{A}}_i$ and tracing over the input. Each ${\mathcal{A}}_i$ maps the output state of the $i\mathrm{th}$ CPTP map to the input leg of the $(i+1)\mathrm{th}$ CPTP map. (e) To reconstruct $\hat{\mathcal{E}}$ experimentally, prepare a complete set of states $\{{\rho }_i\}$, apply $\mathcal{E}$, and reconstruct the output state with an IC POVM $\{{\mathrm{\Pi }}_j\}$. (f) To reconstruct ${\mathrm{{\rm Y}}}_{k:0}$, measure each final state of the system subject to a complete basis of CP maps $\{{\mathcal{B}}_j^{{\mu }_j}\}$ at each time. Note that the blue unitaries here are symbolic of any $SE$ interactions and not gates that need to be performed.

Figure 3

(a) The circuit structure for PTT. An arbitrary state is fed in, the experimenter acts with all combinations of different basis elements at different times, and a final measurement is recorded. (b) The logical flow of PGDB in the context of our MLE-PTT procedure. We maximize the likelihood of the model through iterative gradient descent and the physical projection of Sec. 3a until some convergence condition is achieved. Full details of the algorithm are shown in Appendix pp1.

Figure 4

A comparison between projection methods imposing physical conditions on 500 normally distributed random matrices. (a) The average time taken for a single projection for both Dykstra’s alternating-projection algorithm and the direct conic projection. We compare conditions set by QST, QPT, and PTT. (b) The average number of eigendecompositions for each of the above. This dominates the run time of each method.

Figure 5

The reconstruction fidelity of various three-step process-tensor procedures when using a minimal complete basis. Each data point represents a different randomly generated unitary sequence. The top and bottom of the box plots are 75 and 25 percentiles, the orange line is the median (figure also printed), the whiskers are 1.5 times the interquartile range, and any remaining data points are outliers beyond this. We compare reconstructions with (a) a randomly generated basis and linear inversion, (b) a randomly generated basis processed by maximum likelihood, (c) a mutually unbiased unitary basis (MUUB) processed with linear inversion, and (d) a MUUB processed with maximum likelihood. The results showcase high-fidelity physical process tensors with minimal resources, in contrast to Ref. [21] where the random-basis data are taken.

Figure 6

A contraction strategy for mapping multitime gate sequences using a conditional Markov order ansatz. Here, we show how a four-step process can be modeled by two-step memory process tensors. The memory process tensors are stitched together by first contracting the relevant operations to their times to account for correlations. After tracing over the state output (denoted by $\delta$), the conditionally independent parts can be treated as tensor products and thus stitched together with the latest common operation, where the output of the earlier process tensor is mapped to the input of the later one.

Figure 7

Circuits to construct a process tensor with conditional Markov order $\ell$. For a $k$-step process, ${\mathbf{{\rm Y}}}_{k:0}^{\ell }$, there are $k-\ell +1$ memory process tensors that need to be constructed—each circuit represents the estimation of each of these, by varying all combinations of all indices from $i_0$ to $i_{\ell -1}$. In the other gate positions, a fixed operation ${\mathcal{B}}_f$ is applied.

Figure 8

The build-up of temporal correlations with increasing interaction time. (a) The CQMI quantifies the error in truncating correlations beyond $\ell =2$ when conditioned on intermediate operations. (b) A four-step process is considered, with an increasing wait time after each gate. Using Markov orders $\ell =1$, $\ell =2$, and $\ell =3$, we quantify how each model reconstructs 100 random unitary sequences. The points indicate the average fidelity and the shaded regions indicate the standard deviation. (c) The QMI and CQMI both increase with the interaction time, bounding the breakdown of $\ell =1$ and $\ell =2$ models, respectively.

Figure 9

The results of using the process tensor (PT) to optimize multitime circuits with different random single-qubit unitaries. The $x$ axis indicates the fidelity compared to ideal when the sequences are run on the device. The $y$ axis is the fidelity of the sequence when the process tensor is used to optimize to the ideal case. (a) Here, we have three sequential unitaries with a wait time interleaved similar to that of two cnot gates ($0.71\mu \mathrm{s}$) on the ibmq_manhattan (no background). (b) A similar setup is considered on ibmq_bogota but with the neighboring qubit in a $|+⟩$ state and subject to four cnot gates per time step ($2.5\mu \mathrm{s}$) ($|+⟩$ and cnot background).

Figure 10

The results of using conditional Markov models to improve the fidelities of a five-step circuit on ibmq_montreal and ibmq_guadalupe. We construct conditional Markov order models for $\ell =1$, $\ell =2$, and $\ell =3$ across five steps under a variety of background conditions. For randomized inputs, these models are used to optimize the next four operations. The four optimal operations are then applied to each input and the results compared to the machine fidelity. The mean and standard deviation for each set of circuits is listed, as well as the average reconstruction fidelity of the models. (a) No background operations are applied. (b) The nearest neighbor to the system is initialized in a $|+⟩$ state. (c) Two nearest neighbors and one next-to-nearest neighbor from the system are initialized in a $|+⟩$ state.

Figure 11

PGDB