1 Learning quantum circuits of some T gates Ching-Yi Lai and Hao-Chung Cheng Abstract In this paper, we study the problem of learning quantum circuits of a certain structure. If the unknown target is an n-qubit Clifford circuit, we devise an algorithm to reconstruct its circuit representation by using O(n2) queries to it. It is unknown for decades how to handle circuits beyond the Clifford group for which the stabilizer formalism cannot be applied. Herein, we study quantum circuits of T -depth one given the all-zero state as an input. We show that their output states can be represented by some stabilizer pseudomixtures. By analyzing the algebraic structure of the stabilizer pseudomixture, we can reconstruct the output state of an unknown T -depth one quantum circuit on input j0ni from the outcomes of Pauli measurements and Bell measurements. If the number of T gates is of the order O(log n), our algorithm requires O(n2) queries. Our results greatly extend the previous known facts that stabilizer states can be efficiently identified based on the stabilizer formalism. Hence, the proposed expanded stabilizer formalism and our analysis might pave the way towards learning quantum circuits beyond the Clifford structure. I. INTRODUCTION Consider a quantum circuit U that is accessible to us but its inner working or the mathematical description are unknown. Then, given a quantum state j i as an input, what do we know about the circuit output? The goal of this paper is to find a circuit representation that resembles the functioning of U so that we are able to predict the output Uj i given an arbitrary input j i. Moreover, we would like to use as few queries to U as possible. We term such a problem learning an unknown quantum circuit. If we focus on only the output state Uj i for a particular input state j i, the problem boils down to determining the state Uj i. This is also called quantum state tomography, known as one of the most crucial tasks in quantum information sciences [1], [2], [3], [4]. Its goal is to infer an unknown quantum state (assuming several copies of it are available) through a sequence of quantum measurements in a way so that a proposed candidate state performs well in future predictions. However, this is a non-trivial task. In order to identify an unknown n-qubit quantum state, one would require exponentially many copies (in the number n) of the state during the tomography process to determine a full description of the state. This makes the task of tomography intractable in practice. To mitigate such difficulties, at least two possible approaches were proposed as follows. Firstly, instead of fully characterizing the mathematical description of the unknown state, one might come up with a state that is Probably Approximately Correct (PAC) [5] when only a particular set of measurements is of interest in future predictions. In this case, Aaronson formulated the state tomography as a learning problem, and proved that only O(n) copies of the state are sufficient to obtain a good hypothesis state [6] (see also [7], [8], [9], [10], [11] for the related works). Secondly, one can focus on restricted states with a certain specific structure. For example, stabilizer states are a class of states that play an substantial role in quantum error-correcting codes and other computational tasks [12]. Aaronson and Gottesman provided a procedure to identify an unknown n-qubit stabilizer state with only O(n) copies of it if collective measurements are possible [13]. Later, Montanaro proposed an efficient algorithm via Bell sampling that consumes O(n) copies of the state and runs in time of order O(n3) [14]. Rocchetto cast the problem into the PAC learning model, and showed that stabiliser states are efficiently PAC learnable in the sense that the running time is polynomial in n [15]. (Note that the O(n) number of copies is optimal by Holevo’s theorem [16].) If now one aims to infer an unknown quantum evolution with certain known input states, this is called quantum process tomography. Once we completely know the underlying evolution, we can determine the final states for arbitrary initial states. This is the target problem we want to study in this paper. Nevertheless, this problem is much more challenging than quantum 3n arXiv:2106.12524v1 [quant-ph] 23 Jun 2021 state tomography. The amount of resources needed for identifying an arbitrary n-qubit quantum circuit is 4 , which is also practically formidable [17], [18], [19], [20]. When considering a restricted class of Clifford circuits Cn [21], Low showed that an n-qubit Clifford circuit Cn is determined (up to a global phase) given 2n + 1 queries to Cn and 2n queries to its conjugate y 2 Cn in time O(n ) [22]. Moreover, a converse result showed that at least n queries is required for such the task [22]. However, no concrete algorithms for reconstructing the circuit representation of the target Cn are provided and one is not capable of predicting the output state of Cn when sending an arbitrary state as input. The first main contribution of this paper is to fulfill this gap. Specifically, we propose a constructive algorithm to efficiently produce the circuit representation of the target circuit 2 Cn by using O(n ) queries to it. (See Theorem 9 and Algorithm 1 in Section III.) Theorem 1. [Learning unknown Clifford circuits] Given access to an unknown Clifford circuit Cn, one can learn a circuit 2 3 description using 2n + 10n + 4 queries to it in time O(n ), so that the produced hypothesis circuit is equivalent to Cn with probability at least 1 − 2−n+1. CYL is with the Institute of Communications Engineering, National Yang Ming Chiao Tung University, Hsinchu 30010, Taiwan. email:[email protected] HCC is with the Department of Electrical Engineering, Graduate Institute of Communication Engineering, National Taiwan University (NTU), Taipei 10617, Taiwan, and with the Department of Mathematics, Institute of Applied Mathematical Sciences, NTU, Taipei 10617, Taiwan, and also with Hon Hai (Foxconn) Quantum Computing Centre, New Taipei City 236, Taiwan. 2 y Let us emphasize that our approach does not rely on accessing the conjugate circuit Cn (as it was required in [22]) since implementing such the conjugate circuit might incur exponential overhead [23], hence compromising the efficiency of the learning process. The problem of learning an unknown Clifford circuit is closely related to that of learning stabilizer states since the output of a Clifford on input j0ni is a stabilizer state. In the proposed Algorithm 1, we adopt Montanaro’s Bell sampling algorithm for learning stabilizer states [14] as a subroutine. Moreover, we employ the stabilizer formalism [12], [24], exploiting the desirable structure of the Pauli group to learn the output stabilizers states. Lastly, by showing that a set of evolved Pauli basis can be identified by changing the input basis state appropriately, we determine the circuit representation of the unknown Clifford circuit Cn via a circuit synthesis procedure. In the aforementioned task, we heavily rely on the stabilizer formalism. However, it is unknown for a long time whether one can efficiently identify a quantum state that is produced from a quantum device beyond the class of Clifford circuits. In this work, we aim to provide an algorithm to identify the unknown quantum state output produced from a quantum circuit consisting of Clifford gates and a non-Clifford gate T = j0ih0j+eiπ=4j1ih1j. This Clifford+T gate set is universal for quantum computation and receives great attention in fault-tolerant quantum computation [25], [26], compiling quantum circuits [27], [28], [29], and quantum circuit simulations [30], [31]. Namely, an arbitrary quantum circuit can be approximately decomposed as a sequence of Clifford stages and T stages, alternatively. Here, a Clifford stage is simply a Clifford circuit. In a T stage, either a T gate or the identity is applied to each qubit1. The number of T stages in a circuit is called the T -depth of the circuit. A slightly related work is that the quantum circuits in the Clifford hierarchy can be distinguished by some POVM measurements [22, Theorem 8]. However, no exact construction of the POVMs is given, other than the first level of Clifford hierarchy, namely the Pauli group, which can be identified using the idea of superdense coding [32]. We note that quantum circuits of T -depth one have been studied in [33]. Our second main contribution is as follows. Suppose that U is an unknown n-qubit T -depth one quantum circuit with O(log n) T gates. We show that it requires at most O(n2) queries to efficiently learn the output of U on input j0ni. Theorem 2. [Learning unknown T -depth one output states] Given access to an unknown T -depth one quantum circuit U, one can learn a circuit description using O(3kn) queries to the unknown circuit U with time complexity O(n3 + 3kn), where k ≤ n is the number of T gates, so that the produced hypothesis circuit U^ is equivalent to U with probability at least 1 − 3e−n when the input states are restricted to the computational basis. The explicit procedure is provided in Algorithm 2 of Section V. The reason why learning quantum circuits U of some T gates is a much more technical-demanding problem is elaborated in the following. Since the T gate dose not belong to the Clifford group, Pauli operators are not preserved by the evolution of the quantum circuit U, and hence the stabilizer formalism or the Gottesman–Kitaev theorem [24] does not work for circuits with T gates.