Towards minimal resources of measurement-based quantum computation

We improve the upper bound on the minimal resources required for measurement-only quantum computation (M A Nielsen 2003 Phys. Rev. A 308 96–100; D W Leung 2004 Int. J. Quantum Inform. 2 33; S Perdrix 2005 Int. J. Quantum Inform. 3 219–23). Minimizing the resources required for this model is a key issue for experimental realization of a quantum computer based on projective measurements. This new upper bound also allows one to reply in the negative to the open question presented by Perdrix (2004 Proc. Quantum Communication Measurement and Computing) about the existence of a trade-off between observable and ancillary qubits in measurement-only QC.

of quantum information processing technologies. Moreover, the use of projective measurements instead of unitary transformations opens new theoretical perspectives for the parallelization of quantum algorithms [17].

Measurement-only QC
The simulation of a given unitary transformation U by means of projective measurements can be decomposed into the following.
1. (Simulation step) first, U is probabilistically simulated up to a Pauli operator, leading to σU, where σ is either an identity or a Pauli operator σ x , σ y , or σ z . Each σ occurs with the same probability 1/4 and is known by the classical outcome of the measurements. Since Pauli operators are used to denote both observables and unitary gates, a σ-notation is used to denote the unitary transformations, whereas capital letters are used to denote observables. 2. (Correction) then, a corrective strategy which consists of conditionally combining steps of simulation, is used to obtain a non-probabilistic simulation of U.
Teleportation can be realized by two successive Bell measurements (figure 1), where a Bell measurement is a projective measurement in the basis of the Bell states A step of simulation of U is obtained by replacing the second measurement by a measurement in the basis {(U † ⊗ I)|B ij } i,j∈{0,1} (figure 2).
If a step of simulation is represented as a probabilistic black box (figure 3, left panel), there exists a corrective strategy (figure 3, right panel) which leads to a full simulation of U. This strategy consists of conditionally composing steps of simulation of U, but also of each Pauli operator. A similar simulation step and strategy are given for the two-qubit Step of simulation based on state transfer.
unitary transformation X (controlled-X) in [2]. Notice that this simulation uses four ancillary qubits.
As a consequence, since any unitary transformation can be decomposed into X and onequbit unitary transformations, any unitary transformation can be simulated by means of projective measurements only. More precisely, for any circuit C of size n-with basis X and all one-qubit unitary transformations-and for any > 0, O(n log(n/ )) projective measurements are enough to simulate C with probability greater than 1 − . Approximative universality, based on a finite family of projective measurements, can also be considered. Leung [5] has shown that a family composed of five observables ⊗ X} is approximatively universal, using four ancillary qubits. It means that for any unitary transformation U, any > 0 and any δ > 0, there exists a conditional composition of projective measurements from F 0 leading to the simulation of a unitary transformationŨ with probability greater than 1 − and such that U −Ũ < δ.
In order to decrease the number of two-qubit measurements-four in F 0 -and the number of ancillary qubits, a new scheme called state transfer has been introduced [6]. The state transfer (figure 4) replaces the teleportation scheme for realizing a step of simulation. Composed of one two-qubit measurement, two one-qubit measurements, and using only one ancillary qubit, the state transfer can be used to simulate any one-qubit unitary transformation up to a Pauli operator (figure 5). For instance, simulations of H and HT -see section 3 for definitions of H and T -are Figure 6. Simulation of H and HT up to a Pauli operator. represented in figure 6. Moreover a scheme composed of two two-qubit measurements, two onequbit measurements, and using only one ancillary qubit can be used to simulated X up to a Pauli operator (figure 7). Since {H, T, X} is a universal family of unitary transformations, the family } of observables is approximatively universal, using one ancillary qubit [6]. This result improves the result by Leung, since only one two-qubit measurement and one ancillary qubit are used, instead of four two-qubit measurements and four ancillary qubits. Moreover, one can prove that at least one two-qubit measurement and one ancillary qubit are required for approximative universality. Thus, it turns out that the upper bound on the minimal resources for measurement-only QC differs from the lower bound, on the number of one-qubit measurements only.
In [7], it has been shown that the number of these one-qubit measurements can be decreased, since the family composed of one two-qubit and only two onequbit measurements, is also approximatively universal, using two ancillary qubits. The proof is based on the simulation of X-measurements by means of Z and Z ⊗ X measurements ( figure 8). This result leads to a possible trade-off between the number of one-qubit measurements and the number of ancillary qubits required for approximative universality.
In this paper, we mean to prove that the family F 2 is approximatively universal, using only one ancillary qubit. Thus, the upper bound on the minimal resources required for approximative universality is improved, and moreover we answer the open question of the trade-off between observables and ancillary qubits. Notice that we prove that the trade-off conjectured in [7] does not exist, but another trade-off between observables and ancillary qubits may exist since the bounds on the minimal resources for measurement-only QC are not tight.

Universal family of unitary transformations
There exist several universal families of unitary transformations, for instance {H, T, X} is one of them: We prove that the family {HT, σ y , Z} is also approximatively universal.

Theorem 1. U = {HT, σ y , Z} is approximatively universal.
The proof is based on the following properties (see [18] for details). Let R n (α) be the rotation of the Bloch sphere about the axis n through an angle α. = (a, b, c) is a real unit vector, then for any α, R n (α) = cos(α/2)I − i sin(α/2)(aσ x + bσ y + cσ z ).
U 3 U 1 is a rotation of the Bloch sphere about an axis along m = (− cos(π/8), sin(π/8), cos(π/8)) and through the angle θ. Thus, for any α and any > 0, there exists k such that Since n and m are non-parallel, any one-qubit unitary transformation U, according to proposition 2, can be decomposed into rotations around n and m: there exist α, β, γ, δ such that Finally, for any U and > 0, there exist k 1 , k 2 , k 3 such that Thus, any one-qubit unitary transformation can be approximated by means of U 2 U 1 , and U 3 U 1 . Since U 2 U 1 = (HT)(HT)and U 3 U 1 = σ y HTHσ y T = −(σ y HT)(σ y HT), the family {HT, σ y } approximates any one-qubit unitary transformation.
With the additional Z gate, the family U is approximatively universal.

Universal family of projective measurements
In [7], the family of observables F 2 = {Z ⊗ X, Z, X−Y √ 2 } is proved to be approximatively universal using two ancillary qubits. We prove that this family requires only one ancillary qubit to be universal.
2 } is approximatively universal, using one ancillary qubit only. The proof consists of simulating the unitary transformations of the universal family U. First, one can notice that HT can be simulated up to a Pauli operator, using measurements of F 2 , as depicted in figure 6. So, the universality is reduced to the ability to simulate Z and the Pauli operators-Pauli operators are needed to simulate σ y ∈ U, but also to perform the corrections required by the corrective strategy (figure 3). In order to simulate Pauli operators, a new scheme, different from the state transfer, is introduced. Lemma 6. For a given qubit b and one ancillary qubit a, the sequence of measurements Z a , X a ⊗ Z b and Z a (figure 10) simulates, on qubit b, the application of σ z with probability 1/2 and I with probability 1/2.
Proof. Let | = α|0 + β|1 be the state of qubit b. After the first measurement, the state of the register a, b is |ψ 1 = (σ s 1 x ⊗ I)|0 ⊗ | where s 1 ∈ {0, 1} is the classical outcome of the measurement.
Since ψ 1 |X ⊗ Z|ψ 1 = 0, the state of the register a, b is: Let s 3 ∈ {0, 1} be the outcome of the last measurement, on qubit a. If s 1 = s 3 then state of the qubit b is | , and σ z | otherwise. One can prove that these two possibilities occur with equal probabilities. HT and Z(I ⊗ H) can be simulated up to a Pauli operator (lemma 5). The universality of the family of observables F 2 = {Z ⊗ X, Z, X−Y √ 2 } is reduced to the ability to simulate any Pauli operators. Lemma 7 (resp. lemma 6), shows that σ x (σ z ) can be simulated with probability 1/2, moreover if the simulation fails, the resulting state is same as the original one. Thus, this simulation can be repeated until a full simulation of σ x (σ z ). Finally, σ y = iσ z σ x can be simulated, up to a global phase, by combining simulations of σ x and σ z . Thus, F 2 = {Z ⊗ X, Z, X−Y √ 2 } is approximatively universal using only one ancillary qubit.