Quantum circuits cannot control unknown operations

Quantum circuits cannot control unknown operations ˇ Mateus Araújo,1, 2 Adrien Feix,1, 2 Fabio Costa,1, 2 and Caslav Brukner1, 2 1 Faculty

of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria of Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria (Dated: 1st October 2013) 2 Institute

arXiv:1309.7976v1 [quant-ph] 30 Sep 2013

One of the essential building blocks of classical computer programs is the “if” clause, which executes a subroutine depending on the value of a control variable. Similarly, several quantum algorithms rely on the possibility of applying a quantum operation conditioned on the state of a control system. Here we show that such a control cannot be performed within a quantum circuit if the operation itself is not known in advance. However, when physical operations act on a subspace of an extended Hilbert space, the control becomes possible. This demonstrates that the quantum circuit model must be generalized to account for computational problems in which operations themselves are variables.

Quantum computation harnesses quantum effects to significantly outperform classical computation in solving specific problems. In the most widely used model for quantum computation, a set of quantum systems is prepared in a given state and a set of unitary operations is sequentially applied to them, with each unitary acting on one or more systems. This model – referred to as the quantum circuit model – can be formulated in terms of wires, representing the quantum systems, which connect boxes, representing the unitary operations. The formalism of quantum circuits is often considered as the standard “programming language” for quantum computers. The difference between the classical and the quantum models of computation is not just a matter of computational power: quantum information processing differs in some fundamental, and often counterintuitive, ways from classical computing. One of the most striking examples is the fact that it is impossible to produce a perfect copy of an unknown quantum state [1], whereas copying of classical information is a standard operation for classical computers. Another standard operation in classical computer programs, usually expressed by an “if” clause, is the conditional execution of a part of the program depending on the value of a variable. A typical programming line of this kind has the form “if x = 0, do A”, where A represents an arbitrary set of commands, i.e., a subroutine. Crucially, the construction of the if clause is independent of the subroutine A, allowing the latter to be used as a variable in the program. The quantum analogue of the “if” clause is the control of a unitary operation depending on the value of a control quantum bit (qubit). This is represented by the transformation

(α|0iC + β|1iC )|ψi 7→ α|0iC |ψi + β|1iC U |ψi, where the subscript C stands for the control qubit and

|ψi is the initial state of the “target” system. This “control-U” gate is a fundamental tool for quantum computation. It is used for example in Kitaev’s phase estimation algorithm [2], Shor’s factoring [3], Metropolis sampling [4], and in the DQC1 computing model [5]. The standard strategy to implement this gate in a quantum circuit is to decompose U into elementary gates, for which one knows how to add control [6]. This approach is resource demanding, since it requires O(d2 ) gates to add control to a unitary of dimension d. Most importantly, it requires the unitary to be known, thus it cannot be used for solving problems in which the unitary itself is a variable. A genuine quantum analogue of the if clause would be an implementation of the control-U gate in which U can be treated as a blackbox. Although it has long been suspected that such a construction is impossible (e.g., it is mentioned en passant by Kitaev in Ref. [2]), no proof of this fact is known to the best of our knowledge. Here we prove a no-go theorem, that shows the impossibility of controlling an arbitrary unknown unitary in the quantum circuit model. The formal question is whether a quantum circuit can implement the controlU gate, given as input a single copy of the unknown d-dimensional gate U. Thus we ask whether there exist unitaries A and B such that the following circuit identity is satisfied: |0i

|0i B

A U

?

=

WU

• U

where the topmost line represents an additional adimensional quantum system (ancilla) and WU is an arbitrary unitary on the ancilla, possibly depending on U. Note that the left hand side depicts the most general transformation that a quantum circuit can effect on U

2 [7]. In order to see that the above identity cannot be satisfied, it is sufficient to notice that, in the lhs, substituting U with eiϕ U does not produce any physical difference, since the two circuits only differ by a global phase eiϕ . In contrast, the same substitution in the rhs produces a measurable relative phase. Since the unitary U in the lhs is only defined up to a phase, it is worth asking whether it is possible to implement the control-U modulo this global phase. To translate this question into an equation, note that the action of the control-U operation is given by 1d ⊕ U. Defining |U i a := WU |0i a , the question is whether the identity B(1a ⊗ 12 ⊗ U ) A|0i a = |U i a (1d ⊕ eiu U )

(1)

holds for some arbitrary phase factor eiu . This is still not possible, due to the non-linearity of the transformation U 7→ 1d ⊕ eiu U. This can be readily seen if one assumes that equation (1) is valid for the qubit unitaries X, Z, and H = √1 ( X + Z ), where X and Z are the 2 Pauli matrices. One has then   X+Z B 1a ⊗ 12 ⊗ √ A|0i a = | H i a (12 ⊕ eih H ). 2 Expanding the lhs by linearity and using equation (1) again we get 1 √ | X i a (12 ⊕ eix X )+ 2 1 √ | Z i a (12 ⊕ eiz Z ) = | H i a (12 ⊕ eih H ). 2 Taking the inner product with | H i in the ancilla subsystem, we get the operator equation  1  h H | X i √ 12 ⊕ eix X + 2  1  h H | Z i √ 12 ⊕ eiz Z = 12 ⊕ eih H, 2 which cannot be satisfied for any choice of states | X i, | Z i, | H i and phases eix , eiz , eih . This shows that one cannot control an arbitrary unknown unitary in the quantum circuit model. However, it leaves open the possibility of controlling unitaries belonging to known, specific sets. For example, if one eigenvector of U and its eigenvalue are known, there is a circuit that performs the control [2]. In a similar vein, if one knows that U belongs to a given set of orthogonal unitaries, it is also possible to control it [8]. There is, therefore, a fundamental difference between classical and quantum circuits, since in classical computation control of a blackbox operation is obviously possible. We can represent this possibility in the quantum

circuit formalism if we restrict the target system |ψi to be a classical bit string, i.e., to belong to the computational basis, and the unitary to be a classically allowed transformation Ucl. , i.e. a permutation matrix. Then the following circuit implements the control-Ucl. :

•

|C i |ψi |0i

C

Ucl.

C

where C represents a classical cloning operation (a controlled-NOT for a two-level system). The symbol • means that the operation is applied if the control means that the operation is applied bit is |1i and when the control bit is |0i. Note that if cloning an arbitrary |ψi were possible, the circuit given above would allow one to control unknown quantum operations. This provides an alternative proof of the no-cloning theorem [1]. We will now show that, unlike for the no-cloning theorem, the limitation on controlling a blackbox is a shortcoming of the circuit model of quantum computing, not of quantum mechanics itself. In fact, control of blackbox quantum gates has been experimentally demonstrated [9–11]. Such schemes exploit the possibility of extending computational systems to higher dimensions, but it is still surprising that they do not run in contradiction with the no-go theorem proved above. In order to illustrate how this is possible, we propose here a simple interferometric setup, depicted in Fig. 1, that exploits a similar idea, but implements the control-U operation in a very direct way. A single photon is input in the state (α| H iC + β|V iC )|ψi, where | H i and |V i represent horizontal and vertical polarization states of the photon and |ψi is the state of some other degree of freedom of the same photon (for example, it could be its orbital angular momentum, a qudit of spatial or temporal bins, etc.). Let then U be a unitary gate acting on this additional degree of freedom. It is straightforward to check that the interferometer in Fig. 1 applies the transformation

(α| H iC + β|V iC )|ψi 7→ α| H iC |ψi + β|V iC U |ψi,

(2)

for any blackbox unitary U. So how does this interferometric implementation circumvent the no-go theorem we have just proved? Its essential feature is that it allows one to apply the original d-dimensional gate to a subspace of an extended 2d-dimensional system. In contrast, in a quantum circuit one would only be able to apply a gate to a subsystem of an extended space. To be more explicit, let {|0i, . . . , |d − 1i} be a basis spanning the space in which U acts, and {|ri, |bi} denote the red and blue paths.

3 α|H〉|ψ〉+β |V〉U|ψ〉 β|V〉|ψ〉

(α|H〉 + β|V〉)|ψ〉

U

β|V〉U|ψ〉

PBS

α|H〉|ψ〉

PBS Figure 1. Interferometer that controls a black box U. Here the control qubit is the polarization of the photon, and U acts on some additional degree of freedom of the same photon. The PBSs are polarizing beam splitters. A photon with polarization | H i takes the lower (red) path, while one with polarization |V i takes the upper (blue) path.

Then, in the basis {|ri, |bi} ⊗ {|0i, . . . , |d − 1i}, the operation implemented in the interferometer is represented by   1d 0 Uinterferometer = , 0 U which is exactly the matrix representation of the control-U gate. Thus, given a device that implements U, the interferometer effectively turns it into a device that implements 1d ⊕ U. In short, quantum circuits only allow for extensions of the kind U 7→ 1 ⊗ U, while the extension U 7→ 1d ⊕ U is also physically realizable. Other examples of physical systems in which such an extension can be implemented include time-bins [12], electronic levels of atoms [13] and photonic orbital angular momentum [14]. In general, by exploiting multi-level systems, it is possible to extend a unitary to 1d0 ⊕ U for an arbitrary dimension d0 . Note that a one-dimensional extension is sufficient to implement a control-U using the scheme of Ref. [2]. This is possible because the scheme allows controlling unitaries as long as one eigenvector and the corresponding eigenvalue is known, which is the case for 1 ⊕ U. Our results have profound implications for quantum algorithms in which the quantum operations themselves are used as variables. The framework of quantum circuits has to be extended in this respect, since it only allows one to apply unknown operations on subsystems, while accessing subspaces is necessary to implement the control-U. Taking this into account is paramount to fully appreciate the importance of quantum algorithms that rely on estimating the properties of an unknown operation such as Kitaev’s [2] and DQC1’s trace estimation [5]. Furthermore, the scheme presented could also be used to simplify the imple-

mentation of the control-U gate even when the unitary in question is known. We have shown that a general “programming language” for quantum computing must go beyond quantum circuits, to account for the possibility of the “quantum if”. Other extensions of the quantum circuit formalism have been proposed, in which the wires between gates can be in a superposition [15–17]. Allowing for such “superpositions of circuits” can simplify the implementation of some information processing tasks [16, 17], or even reduce the computational complexity of some problems [18]. It is an intriguing open question whether further extensions are possible. We thank Alessandro Bisio, Jan-Åke Larsson, Amir Moqanaki, Daniel Nagaj, and Michal Sedlák for helpful discussions. This work was supported by the European Commission Project RAQUEL, the John Templeton Foundation, FQXi, and the Austrian Science Fund (FWF) through CoQuS, SFB FoQuS, and the Individual Project 2462.

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