Implementation of Gate Set Tomography on Quantum Hardware
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Implementation of Gate Set Tomography on Quantum Hardware Henrique Guimaraes˜ Silverio´ [email protected] Instituto Superior Tecnico,´ Lisboa, Portugal November 2019 Abstract Finding and understanding errors in quantum devices is instrumental to their development. Among error characterization techniques, Gate Set Tomography (GST) stands as the most comprehensive proto- col. In this work, we implement a working GST protocol to characterize single-qubit operations of selected IBM Q’s quantum devices. By benchmarking GST with simulated data, we find the optimal solutions to be degenerate, with the initial state and readout estimations changing correlatedly. To fix this, we propose a protocol modification which assumes a perfect initial state. The modified GST protocol produces accurate and consistent estimations, with precision limited mainly by sampling noise. In the application of GST to IBM Q devices, we observe that the average single-qubit gate quality is close to the limit of GST’s precision. The same no longer holds for the readout errors, whose larger values agree with IBM’s calibrations. We also characterize a qubit’s readout error over time and find that it often changes in the order of 1% − 2% per hour. Keywords: Quantum computing, GST, qubit characterization, IBM Q 1. Introduction tude of the deviations that appear then speak for In recent years, the field of quantum computation the overall quality of the device. However, a mini- has benefited from substantial interest and invest- mal amount of information can be extracted using ment, which has been translated into the devel- this naive approach. Thus, a more systematic ap- opment of the first widely available prototypes of proach to hardware testing is needed, more specif- quantum computers, with up to 20 qubits. How- ically one that tries to describe the individual ele- ever, these improvements are not yet enough to ments of the device to predict its behaviour on all take full advantage of most of the quantum algo- possible scenarios. rithms under development since the 1990s, in large Qubit characterization provides rigorous tools part due to technological challenges. Further im- for error diagnostics and assessment of the hard- proving current quantum computers relies heavily ware’s performance. The objective is to find out the on understanding the nature and origin of existing sources of errors so that they can be mitigated, or complications, to which extent an accurate method to reliably set an upper bound on the error of an of characterization is essential. The main focus of arbitrary algorithm’s outcome. The different ways this work will be the study, implementation and ap- to approach qubit characterization can be roughly 1 plication to real devices of state of the art charac- sorted by the amount of information they intend to terization techniques. extract from the procedure. Among the more pop- ular methods, Randomized Benchmarking (RB) [7] 1.1. The need for qubit characterization and Gate Set Tomography (GST) [2] stand at op- In the development of quantum devices, testing posite ends of this spectrum. the hardware’s performance is always an essen- tial part of the process. In a first iteration, this can RB is a process in which gate sequences of dif- be done through the computation of quantum al- ferent lengths are applied and the outcome com- gorithms whose outcomes are known; the magni- pared with the predicted value. By doing this, one attains a measurement of the average process fi- 1Namely the IBM Quantum Experience (IBM Q) quantum delity, which can then be used to extract the av- processors. erage gate fidelity [6]. Additionally, this process 1 can be extended to multi-qubit operations with only polynomial scaling with the number of qubits [10]. H In contrast with other techniques, RB is simple, ro- bust and gives reliable average gate fidelities for multi-qubit processes that are useful in ascertain- ing the overall quality of the gates in a quantum State preparation Conditional State device. However, it can only provide the average Manipulation readout fidelity of specific gates, which makes it unsatisfac- Figure 1: Diagram representing the stages of a quantum com- tory as a standalone characterization technique. putation, with the example of a circuitp that prepares the max- imally entangled state (j00i + h11j)= 2. The state prepara- On the other hand, GST tries to achieve a com- tion represents the transformation of an arbitrary quantum state plete characterization of the system. With GST, the onto the j00i state and is usually omitted in quantum circuits. The circuit itself shows the implementation of a Hadamard gate goal is to achieve a description of the hardware that on the first qubit, which creates an equal superposition state on correctly predicts the outcome of any circuit. This that qubit, followed by a CNOT gate that flips the state of the feature makes GST much more informative than second qubit conditionally on the state of the first. In the end, RB, but has the serious drawbacks of being costly each qubit is individually measured in the computational basis, returning one of two possible basis states. in resources and unpractical for multiple qubits. In this work, the method of choice for the charac- terization of IBM’s quantum processors was GST, 2.2. Operations with Density Matrices for several reasons: First and foremost, because In this work, we adopt the density matrix formal- we were interested in achieving a description of the ism, which represents a system as an ensemble of system that was as detailed as possible. Secondly, quantum states with probability pi, fpi; j iig. It is due to the fact that RB has already been exten- defined as an operator sively applied to IBM’s systems, while GST results X remain, to our knowledge, either undone or unpub- ρ = pi j ii h ij (1) lished. Lastly, because the IBM Q is remotely ac- i cessible to the scientific community and has thus been subjected to extensive testing, there are mul- A density matrix, ρ, represents a physical system tiple reports of its poor performance when com- if Tr[ρ] = 1, ρ = ρy and ρ is semi-definite positive pared with simulation results [5]. These reports (i.e. ρ 0) [14]. To describe changes to a system, made clear that understanding what is wrong and we use quantum operations, which are defined as trying to improve the current devices is a pressing maps acting on the system’s density matrix, ρ ! issue, for which purpose GST is a far more useful Λ(ρ). Maps that describe physical processes must tool than RB. meet two basic requirements [6]: 1.2. Objectives 1. Trace preservation: If Tr[ρ] = 1, then Tr[Λ(ρ)] = 1 The principal focus of this work was to implement a working GST protocol capable of fully character- 2. Complete Positivity: For a composite system izing single-qubit operations of present-day quan- ρ 0, then (Λ(·) ⊗ I)ρ 0, where Λ only acts tum devices and testing its performance in both on part of ρ. simulated and real experimental data. Nonethe- less, rather than debugging IBM Q’s systems in For obeying the conditions above, a physical map particular, our main intention was to explore GST’s is often referred to as a CPTP (completely positive general capabilities, to uncover potential limitations trace-preserving) map. In fact, any physical map and to establish the extent of its applicability and can be conveniently written in its Kraus form [6]: usefulness in experimental settings. N X y 2. The principles of GST Λ(ρ) = KiρKi (2) 2.1. Quantum Circuit Model i=1 The computation model used describes quantum where N ≤ d2, with d = 2n being the Hilbert space algorithms as circuits, in which a sequence of dimension of the n-qubit space. The Ki are the quantum gates (i.e. operators) act on qubits (i.e. Kraus operators and they need only fulfill the com- quantum states). Any quantum computation can PN y pleteness condition, i=1 Ki Ki = I. then be divided into three stages: state prepara- tion, conditional manipulation and state readout. 2.3. Superoperator Formalism As an example, figure 1 depicts a typical quantum Despite its generality, the quantum operations for- circuit, subdivided into these three stages. malism lacks the ease of manipulation the Dirac 2 notation offers the state vector formalism. We cir- These definitions lead naturally to (see Section cumvent this issue with the superoperator formal- 2.2 of Ref.[6]): ism, which represents density operators ρ (of di- mension d × d) in the Hilbert space of dimension d jΛ(ρ)ii = RΛjρii (9) as vectors jρii in the Hilbert-Schmidt space of di- R = R R (10) mension d2. In turn, quantum operations now con- Λ1◦Λ2 Λ1 Λ2 sist of d2 ×d2 matrices in this space. We define the Thus, we see that the superoperator formalism Hilbert-Schmidt inner product of two density oper- permits a quantum operation acting on a density ators A and B as matrix to be represented as a PTM acting on a state vector. A sequence of quantum operations y hhAjBii = Tr[A B] (3) can also be represented as a simple multiplication of the respective PTMs. This formalism also supports the representation of some observable, E, since Physicality Constraints hEi = Tr[Eρ] = hhEjρii (4) A physical quantum operation maps a density ma- trix ρ to another density matrix Λ(ρ). This im- Therefore, a generic observable E is represented plies that Λ(ρ) can be decomposed in the normal- hhEj as in the superoperator formalism. ized Paulip basisp with real coefficients in the inter- val [−1= d; 1= d]. Considering the PTM’s param- Choice of basis eters definition in eq.