Implementation of a general single-qubit positive operator-valued measure on a circuit-based quantum computer Yordan S. Yordanov and Crispin H. W. Barnes Cavendish Laboratory, Department of Physics, University of Cambridge, Cambridge CB3 0HE, United Kingdom (Dated: January 15, 2020) We derive a deterministic protocol to implement a general single-qubit POVM on near-term circuit-based quantum computers. The protocol has a modular structure, such that an n-element POVM is implemented as a sequence of (n − 1) circuit modules. Each module performs a 2-element POVM. Two variations of the protocol are suggested, one optimal in terms of number of ancilla qubits, the other optimal in terms of number of qubit gate operations and quantum circuit depth. We use the protocol to implement 2- and 3-element POVMs on two publicly available quantum computing devices. The results we obtain suggest that implementing non-trivial POVMs could be within the reach of the current noisy quantum computing devices. I. INTRODUCTION of the protocol, in terms of number of quantum gates, is 2 O(n ) using log2 n ancilla qubits, and can be reduced to O(n log n)d at a coste of ( log n 1) additional an- In quantum mechanics positive operator-valued mea- 2 cilla qubits. The correspondingd circuite − depths are O(n2) sures (POVMs) describe the most general form of quan- and O(n) respectively. We use the protocol to imple- tum measurement. They are able to distinguish prob- ment 2- and 3-element POVMs on two public quantum abilistically between non-orthogonal quantum states [1] computing devices; IBMQX2 and Aspen4. We measure and can therefore be used to perform optimal state the output fidelities and compare the performances of the discrimination [2,3] and efficient quantum tomography two devices. [4,5]. In quantum communication and cryptography[6], In sec. II we present our protocol. We describe ex- they are used to enable secure device independent com- plicitly how to construct a quantum-gate circuit for a 2- munication [7], or, on the contrary, compromise quan- element POVM, and demonstrate how it can be extended tum key distribution protocols by minimizing the damage to a n-element POVM. In sec. III we present the results done by an eavesdropper to a quantum channel [8,9]. from the POVM implementations on the two quantum POVMs can be implemented experimentally in both devices. We present our concluding remarks in sec.IV. bosonic [10{13] and fermionic quantum systems [14]. However, typically, the hardware for these implementa- tions needs to be specifically tailored to the measurement. II. POVM PROTOCOL To realize an arbitrary POVM as part of quantum com- munication scheme or on a quantum computer, where the Preliminaries: An n-element POVM is defined as a hardware design allows only orthogonal projective mea- set of n positive operators E^ that satisfy the complete- surements in the qubit basis, it is necessary to simulate i Pn ^ ^f g ^ ^ ^ the action of the POVM using quantum-gate operations. ness relation i=1 Ei = I, where Ei = MiyMi and the ^ For example, in reference [15] a quantum Fourier trans- Mi are measurement operators. Performing a POVM f g form is used to implement a restricted class of projective on a system in initial state 0 results in wave func- j i POVMs. In references [16, 17] a probabilistic method, tion reduction to one of n possible measurement out- ^ based on classical randomness and post-selection, is pro- Mi 0 comes 0 i = j y i , with probability p 0 M^ M^ i 0 posed to implement projective POVMs. A deterministic j i ! j i h j i j i p = M^ yM^ . Using Neumark's theorem, a n- method to perform a general POVM can be implemented i h 0j i ij 0i using Neumark's dilation theorem[18, 19], which states element POVM on a target system A, can be performed arXiv:2001.04749v1 [quant-ph] 14 Jan 2020 that a POVM of n elements can be performed as a pro- by introducing an ancilla system B, with Hilbert space spanned by n orthonormal basis states i(B) that are jective measurement in a n-dimensional space. In refer- j i ence [20] it is shown that this method can be realized in in one-to-one correspondence with the POVM measure- ^ a duality quantum computer. ment outcomes. A unitary operation UAB is applied to the joint state of the two systems, such that In this work we construct a protocol for a general single-qubit POVM on a circuit-based quantum com- n X puter, using Neumark's theorem. The protocol has a U^ (A) 0(B) = M^ (A) i(B) : (1) ABj 0 ij i ij 0 i j i modular structure such that a quantum circuit for a n- i=1 element POVM is constructed as a sequence of (n 1) 2-element POVM circuit modules, in a similar manner− to By performing a projective measurement on system B, ^ (A) reference [10]. This structure allows for a straightforward system A collapses to one of the n states Mi 0 that construction of quantum circuits, using an optimal num- correspond to the outcomes of the POVM. Forj morei de- ber of ancilla qubits and quantum gates. The complexity tails on POVM implementation refer to [21, 22]. 2 Protocol outline: Based on the method, described operations V^1 and V^2 on the terms in the target qubit above, we implement a n-element POVM on a target sys- state, corresponding to the two outcomes of the POVM. tem consisting of a single qubit, using an ancilla system This can be done by two single-qubit unitary gates acting ^ of log2 n qubits. To implement UAB, we divide it into a on the target qubit, and controlled by the ancilla states sequenced e of (n 1) quantum gate circuits, which we call corresponding to the two POVM outcomes, 0 and 1 modules. Each− of these modules, except the first, per- respectively. This results in a final state j i j i forms a 2-element POVM on one of the outcomes of the preceding module, and entangles the additionally pro- Ψ Ψf = V^1D^ 1U^ 0 0 + V^2D^ 2U^ 0 1 ; (6) duced outcome to a new state of the ancilla system. j i ! j i j i j i j i j i with V^1D^ 1U^ = M^ 1 and V^1D^ 2U^ = M^ 2. Since U^, V^1 and V^2 ^ ^ ^ ^ are unitaries, and D1D1y+D2D2y = I, it is straightforward ψ0 U V1 V2 to check that M^ and M^ satisfy the completeness rela- | i • 1 2 tion. Furthermore, the expressions for the two measure- ment operators are in most general form, since they cor- 0 Ry(θ1) Ry(θ2) | i • respond to singular value decompositions. Therefore eq. (6) corresponds to the outcomes of a general 2-element FIG. 1: A quantum circuit for a general single-qubit POVM. Figure1 illustrates the complete circuit for the 2-element POVM. The top qubit acts as the target, and 2-element POVM module. the bottom as the ancilla. The output state of the Generalization to n-element POVM A n-element circuit is given by eq. (6). R^y(θ) denotes a controlled POVM can be performed sequentially by (n 1) POVM − single-qubit y-rotation by angle θ. U^, V^1 and V^2 denote modules, that share an ancilla register of log2 n qubits. th d e general single-qubit unitary operations. V^1 and V^2 are The i module in the sequence will be characterized by (i) (i) ^ (i) controlled operations, and each of them can be rotation angles θ1 and θ2 , unitary operations V1 and implemented as a combination of controlled z and ^ (i) − V2 , and two POVM outcomes with corresponding or- y rotations. The circuit contains up to 12 CNOTs and (i) (i) − thogonal ancilla register states o and o . The first 14 single-qubit rotations. j 1 i j 1 i module is additionally characterized by the unitary U^ acting on the target qubit, as shown above. Each of the modules, except the first one, performs a 2-element 2-element POVM module: To construct a quan- POVM on the second outcome of the preceding module, tum circuit performing a 2-element POVM, we need a so that the term in the target qubit state, corresponding single ancilla qubit. We assume the target qubit starts to this outcome, is evolved in a similar way as for the in an arbitrary state = a 0 + b 1 . Then the initial j 0i j i j i case of the 2-element POVM. The output state of the state of the system, target plus ancilla, is Ψ0 = 0 0 . sequence of modules can be written as To perform a 2-element POVM we want toj transformi j ij thei n 1 system to a state − X ^ (i) ^ (n 1) Ψ = Mi 0 o1 + Mn 0 o2 − (7) Ψf = M^ 1 0 ) o1 + M^ 2 0 ) o2 ; (2) j i j i j i j i j i j i j i j i j i j i i=1 where M^ 1 and M^ 2 are the two measurement operators with the measurement operators M^ i given by and o1 and o2 are two orthogonal states of the ancilla. j i j i ^ 8 First a unitary gate U (not to be confused with UAB) is ^ ^ (1) ^ (1) >M1 = V1 D1 , for i =1 performed on the initial state of the target qubit: > > ^ > Ψ0 U 0 0 = a0 0 + b0 1 0 : (3) < ^ (i) ^ (i) Qi 1 ^ (j) ^ (j) ^ j i ! j i j i j i j i j i M^ i = V1 D1 j−=1 V2 D2 U , for 1 < i < n Then, two controlled y-rotations are performed, acting on > the ancilla qubit and controlled by the target qubit.