Universal quantum computation via quantum controlled classical operations

Sebastian Horvat,1, ∗ Xiaoqin Gao,1, 2, 3, † and Borivoje Dakic1, 2, ‡

1University of Vienna, Faculty of Physics, Vienna Center for QuantumScience and Technology, Boltzmanngasse 5, 1090 Vienna, Austria

2Institute for Quantum Optics and Quantum Information (IQOQI),Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria.

3Department of physics, University of Ottawa, Advanced Research Complex,25 Templeton Street, K1N 6N5, Ottawa, ON, Canada

(Dated: February 11, 2022)

A universal set of gates for (classical or quantum) computation is a set of gates that can be used toapproximate any other operation. It is well known that a universal set for classical computation aug-mented with the Hadamard gate results in universal quantum computing. Motivated by the latter,we pose the following question: can one perform universal quantum computation by supplementinga set of classical gates with a quantum control, and a set of quantum gates operating solely on thelatter? In this work we provide an affirmative answer to this question by considering a computationalmodel that consists of 2n target bits together with a set of classical gates controlled by log(2n + 1)ancillary qubits. We show that this model is equivalent to a quantum computer operating on nqubits. Furthermore, we show that even a primitive computer that is capable of implementing onlySWAP gates, can be lifted to universal quantum computing, if aided with an appropriate quantumcontrol of logarithmic size. Our results thus exemplify the information processing power broughtforth by the quantum control system.

INTRODUCTION

Universal sets of gates for quantum computation arediscrete sets of elementary unitary transformations whichcan be used to approximate any other unitary operation.Starting with Deutsch’s seminal proof of the universal-ity of three-qubit gates [1], the search for universal setsof gates has been established as a mature theme in thefield of quantum computing, and a plethora of variousother universal sets have been thereafter constructed [2–6]. Universal sets can be analyzed from two complemen-tary perspectives: a practical one, which focuses on theefficiency and feasibility of implementing the gates, withthe aim of providing a firm basis for future quantum com-puters; and a more conceptual one, which investigateshow various universal sets shed light on the qualitativedifferences between classical and quantum computation.The latter perspective is exemplified in the works of Shiand Aharonov [7, 8], which show that the Toffoli gate,when supplemented with the Hadamard gate, constitutesa universal set: as the Toffoli gate is universal for classicalcomputation, one can see this result as highlighting theimportance of the Hadamard gate (or, more generally, ofthe quantum Fourier transform) for the computationaladvantage brought forth by quantum computation.

Generally, if one wants to implement a unitary trans-formation on a target system, one can either act di-rectly on that system (e.g. with a unitary device), orone can implement it indirectly by performing transfor-mations and measurements on an ancillary control sys-

∗ sebastian.horvat@univie.ac.at† xgao5@uottawa.ca‡ borivoje.dakic@univie.ac.at

tem. The latter method has been extensively studied inthe context of quantum steering [9], measurement basedquantum computing [10–12], fusion-based quantum com-puting [13, 14] and ancilla driven quantum computation(ADQC) [15–18]. Motivated by the aforementioned re-sults obtained by Shi and Aharonov, we may now askthe following question: can one achieve universal quan-tum computation by supplementing classical operationswith a quantum control, and with a set of quantum gatesacting solely on the latter? More precisely, our aim isto consider computational models composed of classicalsets of gates acting on a target system and an ancillarycontrol system on which unitaries and measurements canbe applied, and inspect whether they are sufficient foruniversal quantum computing.

In this work we answer affirmatively to our questionby constructing a computational model that consists of(i) a set of classical gates acting on 2n target bits, (ii)log(2n+ 1) ancillary qubits which can coherently controlthe classical gates, and (iii) a set of quantum gates avail-able on the control system. We show that this model canbe used to perform universal quantum computation onn qubits which are encoded in a subspace of the Hilbertspace built upon the 2n target bits. The computation isexecuted via unitary transformations and measurementsapplied on the control system in a repeat-until-successmanner [19–21]. Interestingly, the set of classical gatesthat we start with consists only of local NOT and CNOTgates, which can be implemented by the so called paritycomputer, thereby exhibiting a parallel between our workand the study of the “computational power of correla-tions” [22]. Namely, in the latter work the authors provedthat a parity computer can be used to perform univer-sal classical computation, if supplemented with three-qubit GHZ states. On the other hand, in our work we

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show that a parity computer, when supplemented withan appropriate quantum control, can perform universalquantum computation. As an addition, we show that theclassical operations that we use can be alternatively im-plemented as local SWAP-gates, which implies that evena computer that can implement only such “trivial class”of reversible classical operations [23], can be raised touniversal quantum computation.

Returning back to the parallel with Shi’s andAharonov’s results, our work shows an alternativemethod of turning a set of classical gates into a uni-versal set for quantum computation: one simply needsto have access to an ancillary control degree of freedomon which certain unitary transformations (not necessar-ily the whole unitary group) can be implemented. Con-sequently, our results can be seen as a contribution torecent developments that show the information-theoreticadvantage brought forth by the possibility of coherentlycontrolling (quantum) operations, such as the coherentcontrol of orders [24], directions [25], communicationchannels [26–29], and tasks and addressing in quantumnetworks [30].

DESCRIPTION OF THE MODEL

The aim of this work is to construct a model of quan-tum computation via quantum controlled classical oper-ations. The model will consist of a “control system” andof a “target system”, where the former controls whichoperations are to be applied on the latter. In this sectionwe will describe our model and introduce the main con-cepts, whereas the proof of universality will be given inthe subsequent section.

Classical operations – In this subsection we will statethe precise meaning of “classical operations”. Let us as-sume that the target system is a 2n-dimensional quantumsystem, i.e. a collection of n qubits with its Hilbert spacespanned by states {|~x〉 ≡ |x1...xn〉 ,∀xi = 0, 1}; the latterbasis will be referred to as the classical basis. Moreover,let us assume that instead of having available the fullunitary group U(2n) acting on the n qubits, we have atdisposal only a finite subset G ⊂ U(2n). We will saythat the operations Gi ∈ G are classical if they are allequivalent to permutation matrices in the classical basis,i.e. if G ⊆ S2n , where S2n is the symmetric group on2n elements (we are thus considering only reversible clas-sical transformations). For example, for a single qubit(n = 1), the only existing classical operations are theidentity 1 and the NOT-gate, which act as 1 |x〉 = |x〉and NOT|x〉 = |x⊕ 1〉. The reason we call the trans-formations pertaining to G classical is that by restrict-ing the set of available transformations to G, our tar-get system behaves effectively as a classical system, i.e.the n qubits behave effectively as n bits1. More pre-

1 In other words, the limitation on the available transformations

cisely, each state |~x〉 of the n qubits can be regarded asa state ~x of n bits, and each operator Gi ∈ G acting on

the qubits corresponds to a reversible function ~f (i) (~x)

of the n bits, where |~f (i) (~x)〉 = Gi |~x〉. Additionally,all projective measurements in the classical basis corre-spond to “classical measurements” of the bits. Therefore,our system, together with the set of classical operationsG ⊆ S2n , can be used only to perform classical com-putation (if the set G is rich enough, one may be ableto do universal classical computation on n bits or ona subset thereof). Due to the above, we will through-out the manuscript interchangeably use the words “bits”and “qubits” when referring to the target system. Onemay generalize the so far introduced definitions to mixedstates and to generic completely positive and trace pre-serving (CPTP) maps, where classical operations wouldcorrespond to stochastic matrices (and would thus alsoinclude irreversible transformations); however, in thiswork we will focus only on pure states and on unitarytransformations/permutations, in order to avoid unnec-essary complications.Control system – We here introduce the control sys-

tem as a |G|-dimensional quantum system with its Hilbertspace spanned by states {|Gi〉 ,∀Gi ∈ G}; the latter basisof the control system will be referred to as the computa-tional basis. The role of the control system is to controlthe operations G that act on the target system, i.e. theinteraction between the control and target systems is rep-resented as:

|Gi〉 ⊗ |~x〉 → |Gi〉 ⊗Gi |~x〉 . (1)

As opposed to the target system, we will not imposeany restriction on the availability of operations acting onthe control system, and will thus assume that any unitaryoperator can be implemented on it.

Now we briefly show a way in which the control systemcan be used to implement transformations on the targetsystem, that lie outside of the pregiven classical set G.Let us suppose that the initial (control+target) systemis in state |G0〉 ⊗ |~x〉, for arbitrary G0 ∈ G and arbitrary~x. The general scheme, which is illustrated in Figure 1is partitioned in the following three steps.(a) First, we start by acting with a unitary transfor-

mation U (a) on the control system, and we then let thecontrol and target systems interact, leading to:

→∑i

u(a)i0 |Gi〉 ⊗Gi |~x〉 , (2)

where u(a)i0 = 〈Gi|U (a) |G0〉.

(b) Next, we apply a unitary operator U (b) on thecontrol system, resulting in:

→∑j

|Gj〉 ⊗O(ab)j |~x〉 , (3)

superselects quantum theory on n qubits into an effective classi-cal theory on n bits.

3

𝒢

𝑈(𝑎) 𝑈(𝑏)

feedback

𝐶

𝑇

FIG. 1. The figure represents the general scheme for the im-plementation of transformations that lie outside of the classi-cal set G. The classical operations acting on the target systemT are coherently controlled by the control system C. U (a) andU (b) are generic unitaries acting on the control system. Thearrows signify potential iterations of the procedure.

where O(ab)j ≡

∑i u

(b)ji u

(a)i0 Gi.

(c) Finally, we perform a projective measurement{|Gj〉 〈Gj | ,∀j} on the control system. For each outcome“j” the target system is thus postselected (up to normal-

ization) into state O(ab)j |~x〉.

Therefore, the scheme (a)-(c) enables the probabilistic

implementation of target system transformations O(ab)j ,

which may lie outside of G. By reiterating this procedureone can probabilistically implement products of trans-

formations O(ab)j , thereby forming the semigroup L [G],

which is defined as:

L [G] ≡< O(ab)j ,∀U (a,b) ∈ U(|G|); j = 1, ..., |G| >, (4)

where < S > indicates the semigroup generated by set

S, and O(ab)j =

∑i u

(b)ji u

(a)i0 Gi. We will say that the set G

can be lifted to a (semi)group G with the aid of a control

system, if G ⊆ L [G], that is, every element g ∈ G can beimplemented by iterating procedure (a)-(c), in a repeat-until-success manner [19–21]. Notice that the notion of

lifting a set G to the (semi)group G can in principle bedefined for any subset G ⊆ U(2n); in this work we arehowever explicitly interested in G being a set of classicaltransformations, i.e. G ⊆ S2n .

Recall that our goal here is to show the possibility ofachieving universal quantum computation via quantumcontrolled classical operations. Now we are finally readyto formalize the latter problem in terms of the followingquestion: can we find a set of classical transformationsG ⊆ S2n that can be lifted to the k-qubit unitary group,i.e. U(2k) ⊆ L [G], for some k ∈ N? Moreover, how largedoes k need to be, i.e. is it possible to achieve linearscaling k = O(n)? In the next paragraph we will providesome preliminary steps in the analysis of this problem.

Preliminary steps – Our goal is to find a classical setG which can be lifted to the unitary group U(2k) actingon k qubits with the aid of a control system. Notice thatin order to show that U(2k) ⊆ L [G], it is sufficient toshow that L [G] contains any universal set of gates. Theuniversal set that we are going to consider is the standardset of gates {H,T,CNOT} [31], where H and T are theHadamard and phase gate, which in the computationalbasis read as

H =1√2

(1 11 −1

), T =

(1 00 eiπ/4

), (5)

and CNOT is a two-qubit gate which acts asCNOT |x1x2〉 = |x1(x2 ⊕ x1)〉.

We will start with the attempt of constructing thesingle-qubit gates H and T within our model (whichwould enable the construction of any other U(2) trans-formation). Let us tentatively take the target systemto be a single qubit, with the set of available classicaloperations being the maximal one, i.e. G = {1,NOT}.As there are only two available transformations, the con-trol system is two-dimensional, and thus also a qubit.By iterating procedure (a)-(c) described in the previousparagraph, one can implement only transformations ofthe form ∼ (β11 + β2NOT) on the target system, whereβ1,2 are accordingly normalized coefficients. Notice thatall of these transformations mutually commute, whichimmediately points to the impossibility of implementingthe required gates H and T , as the latter do not com-mute. Therefore, if the target system is a single qubit,the set of classical transformations cannot be lifted to thefull group U(2). Nevertheless, it may be possible to use atarget system that consists of more qubits, say k of them,with the available classical set being G ⊆ S2k , and to con-struct the single-qubit group U(2) in a two-dimensionalsubspace of the k qubits’ Hilbert space. Formally, in or-der to see this possibility, note that the (maximal) groupof classical operations Sd is represented on the target’sHilbert space as the group of d×d permutation matrices,thus it can be decomposed into the trivial 1-dimensionalirreducible representation and its complementary (d−1)-dimensional irreducible representation, the latter beingknown as the “standard representation” [32]. The Burn-side theorem then implies that the standard represen-tation spans the set of all (d − 1)-dimensional matrices[33]. Consequently, if the target system consists of kqubits, together with the set of available transformationsbeing G = S2k , we expect the possibility of implement-ing the unitary group U(2k − 1) on the correspondingirreducible subspace of the k-qubits’ space. This impliesthat if we consider the target system to be composed oftwo qubits, together with the set of available operationsbeing G = S3 ⊆ S4, we expect the possibility of buildingany unitary transformation U(2) on a two-dimensionalsubspace of the two-qubit Hilbert space. The subspacein which the constructed U(2) acts will be called the logi-cal subspace, whereas the system that corresponds to thissubspace will be called the logical qubit. In what follows,

4

we will fix the classical set G and specify the logical sub-space.

Fixing the set G and encoding the logical qubits – Asargued in the previous paragraph, we need to take atleast two target qubits and encode the logical qubit intoa two-dimensional subspace. We will choose the set ofclassical operations to be the following set of single-qubitand two-qubit classical gates:

G = {1,NOT1,CNOT} , (6)

where NOT1 implements the NOT-gate on the first qubit,i.e. NOT1 |x1x2〉 = |(x1 ⊕ 1)x2〉, and the CNOT-gateacts as CNOT |x1x2〉 = |x1(x2 ⊕ x1)〉. Notice that theNOT and CNOT gates do not commute, which hints tothe possibility of implementing the group U(2) in a cer-tain subspace. Throughout the letter, we will for sim-plicity use the notation G1 = NOT1, and G2 = CNOT.

Now we will specify the logical subspace, i.e. the sub-space in which the logical qubit will be encoded. Wechoose the computational basis {|0〉L , |1〉L} of the logi-cal qubit to be encoded as:

|0〉L ≡1√2

(|10〉 − |11〉) ,

|1〉L ≡1√2

(|00〉 − |01〉) .(7)

By short inspection one can see that the classical trans-formations G1 and G2 act on the computational statesof the logical qubit as follows:

G1 |y〉L = |y ⊕ 1〉L ,G2 |y〉L = −(−1)y |y〉L ,

(8)

for y = 0, 1. Therefore, the transformations act onthe logical qubit as G1

∼= X, and G2∼= −Z, where

X and Z are Pauli operators. Since the Pauli-Y op-erator can be straightforwardly obtained as G2G1

∼=−iY , we see that due to the particular way of encod-ing the logical qubit, the classical operations G fur-nish automatically the whole Pauli group, up to globalphases. More precisely, the Pauli group for a sin-gle qubit is P =

{eiθ

π2 σj | θ, j = 0, 1, 2, 3

}, where

σ = (1, X, Y, Z). Next, consider the quotient of Pwith its subgroup Z4

∼={eiθ

π2 1| θ = 0, 1, 2, 3

}, which

reads P/Z4 = {[σj ] | j = 0, 1, 2, 3}, where [σj ] ={eiθ

π2 σj , θ = 0, 1, 2, 3

}. From the above considerations

it follows that the set of classical operations G acting on2 qubits furnishes automatically the group P/Z4 actingon the logical qubit.

We will analogously construct n logical qubits from2n target ones, and assume the availability of the trans-

formations G on each pair. Let us label with G(l)i the

transformation that acts as Gi on the l-th target pair(2l − 1, 2l), and trivially on the rest of the qubits, i.e.

G(l)i ≡ 1⊗(2l−2) ⊗Gi ⊗ 1⊗(2n−2l). The total set of avail-

able transformations is then:

G =

n⋃l=1

{G

(l)1 , G

(l)2

}∪ {1} . (9)

𝒢 1

𝒢 2

𝒢 𝑛

𝑀𝑖−1 𝑀𝑖𝐶

𝑥1

𝑥3𝑥4

𝑥2𝑛−1𝑥2𝑛

𝑥2

𝑀1 𝑀2 𝑀3

FIG. 2. Our model of computation which consists of 2n tar-get qubits and a (2n+ 1)-dimensional control system C. Theclassical operations G are local on each qubit pair, and arecoherently controlled by C. The operations Mi represent uni-tary transformations and/or projective measurements appliedon the control system.

Since the latter set contains (2n + 1) elements, thecontrol system has to be (2n+1)-dimensional, or in otherwords, a collection of log(2n+1) qubits. As we will see inthe next section, in order to lift the set G to the unitarygroup U(2n), we will not need to assume the availabilityof the whole group U(2n+1) acting on the control system,but only of a subgroup of the latter.

To recapitulate, our computational model, which ispictured in Figure 2, consists of: (i) the target systemthat consists of n qubit pairs, (ii) the set of classicaltransformations G acting on the target system, (iii) a(2n+1)-dimensional control system, and (iv) a subgroupof U(2n+ 1) available on the control system. In the nextsection we will show that this model can be used to per-form universal quantum computation.

UNIVERSAL QUANTUM COMPUTATION

In this section we will show that the model introducedin the previous section admits the implementation of thestandard set of gates and thus enables universal quantumcomputation. Each gate will be implemented by iteratingthe scheme (a)-(c) presented in the previous section, andillustrated in Figure 1. However, before proceeding withthe implementation of the elementary gates, we need toshow how to initialize the 2n target qubits into a logi-cal qubits’ state, say |0〉⊗nL , that is henceforth ready forcomputation.

Initialization – Let us assume that each pair of qubitsis prepared in the classical state |10〉, i.e. that the joint

state of the target is initially |10〉⊗n. We will now takeone of the pairs (for simplicity, we will omit the pairindex) and show how to initialize it to state |0〉L (theprocedure for the rest of the qubits then follows analo-gously).

Let us take the control system to be initially prepared

5

in state |1〉. The initialization is partitioned in threesteps as follows:

(a) We apply a Hadamard gate on the control systemin the {|1〉 , |G2〉} subspace in order to prepare it in state1√2(|1〉+ |G2〉). We then let the control and target inter-

act, which leads to:

→ 1√2

(|1〉 ⊗ 1 |10〉+ |G2〉 ⊗G2 |10〉)

=1√2

(|1〉 ⊗ |10〉+ |G2〉 ⊗ |11〉) .(10)

(b) Once again, we apply a Hadamard gate (in thesame subspace as in the previous step) on the controlsystem, resulting in:

→ 1

2[|1〉 ⊗ (|10〉+ |11〉) + |G2〉 ⊗ (|10〉 − |11〉)] . (11)

(c) Finally, we perform a projective measurement onthe control system in the computational basis:

• if the outcome is “G2”, then the postselected stateof the target system is 1√

2(|10〉 − |11〉) = |0〉L and

the initialization is complete;

• if the outcome is “1”, the state of the targetsystem is 1√

2(|10〉+ |11〉). We then perform a

“classical measurement” on the target system,i.e. a projective measurement with projectors{|x1x2〉 〈x1x2| ,∀x1, x2 = 0, 1}, which sets the tar-get system either in state |10〉 or in state |11〉. Re-gardless of the outcome of the latter measurement,we reiterate the whole procedure (a)-(c).

We thus keep reiterating points (a)-(c) until the mea-surement in step (c) outputs “G2”; it is straightforwardto see that the target system is then set in state |0〉Lor − |0〉L and the initialization is thus complete, as theglobal phase is irrelevant. The probability of achievingthe desired outcome in m runs grows exponentially fast

towards certainty and is equal to P (m) =∑mi=1

(12

)i=

1 −(12

)m. We can analogously initialize n qubit pairs

which are prepared in the classical state |10〉⊗n, into the

logical state |0〉⊗n; the probability of successfully doing

the latter within m iterations per qubit is(1−

(12

)m)n.

Single-qubit gates – We will now start by showing howto implement the single-qubit gates H and T . We willthus again take one of the qubit pairs and omit the pairindex for simplicity.

H gate. We are going to show how to implement theHadamard gate H on a single logical qubit. Let us as-sume that the initial state of the joint (control+target)system is |G1〉 ⊗ |ψ〉L, where |ψ〉L is an arbitrary purestate pertaining to the logical subspace of the target sys-tem. The procedure consists of the following steps:

(a) First, notice that H = 1√2

(X + Z). Since we know

that the operations G1 and G2 are equivalent to the Pauli

operators X and −Z in the logical subspace, this moti-vates us to apply a Hadamard gate on the control systemin the {|G1〉 , |G2〉} subspace. The subsequent interactionbetween the control and target then leads to:

→ 1√2

(|G1〉 ⊗X |ψ〉L − |G2〉 ⊗ Z |ψ〉L) . (12)

(b) We proceed by again applying the Hadamard gate(in the same subspace as in step (a)) on the control sys-tem:

→ 1

2(|G1〉 ⊗ (X − Z) |ψ〉L + |G2〉 ⊗ (X + Z) |ψ〉L) .

(13)(c) Finally, we measure the control system in the com-

putational basis: if the outcome is “G1”, the target sys-tem is in state 1√

2(X − Z) |ψ〉L = H |ψ〉L, where we de-

fined the unitary transformation H ≡ 1√2(X − Z); on

the other hand, if the outcome is “G2”, the target sys-tem is in state 1√

2(X +Z) |ψ〉L = H |ψ〉L and the proce-

dure is terminated. Notice that the transformations Hand H can be obtained from the Pauli operators X andZ via a π/2-rotation around the y-axis, which impliesthat they are elements of the Pauli group (up to global

phases), i.e. GH ≡{

[1], [H], [H], [Y ]}∼= P/Z4. This

means that for the “wrong” outcome “G1” we can reiter-ate the same procedure (a)-(c) and obtain the target sys-

tem in state HH |ψ〉L or HH |ψ〉L, where HH = 1 ∈ GHand HH = iY ∈ GH . We can continue reiterating theprocedure and after an exponentially short number oftrials the desired gate H is obtained. In the Supple-mentary Information we show that the probability of im-plementing the required gate H within m iterations is

P (m) = 1−(12

)dm/2e.

T-gate. The implementation of the T -gate is analo-gous to the implementation of the Hadamard gate andproceeds as follows. Let us assume that the initial stateof the whole system is |1〉⊗ |ψ〉L, where |ψ〉L is again anarbitrary logical qubit state. As before, the procedure ispartitioned in three steps.

(a) Note that T = eiπ/8 (cos(π/8)1− i sin(π/8)Z).This motivates us to act on the control systemwith the unitary transformation HTH, where Hand T are the Hadamard gate and T -gate in the{|1〉 , |G2〉} subspace, thereby setting the control instate eiπ/8 (cos(π/8) |1〉 − i sin(π/8) |G2〉). The subse-quent control-target interaction then leads to the follow-ing state (up to a global phase):

→ cos(π/8) |1〉⊗1 |ψ〉L−i sin(π/8) |G2〉⊗G2 |ψ〉L . (14)

(b) Next, we apply a Hadamard gate on the controlsystem (in the same subspace as in the previous step),thereby obtaining:

(|1〉 ⊗ (cos(π/8)1− i sin(π/8)G2) |ψ〉L +

+ |G2〉 ⊗ (cos(π/8)1 + i sin(π/8)G2) |ψ〉L).(15)

6

𝒢 𝑘

𝒢 𝑙

𝐻𝐶

𝑥2𝑘−1

𝑥2𝑙−1𝑥2𝑙

𝑥2𝑘

𝐻 𝐻 𝑘 → 𝑙

feedback

FIG. 3. Implementation of the CZ gate acting on the k-thand l-th logical qubits. The gate labelled with “k → l” stands

for the unitary transformation that acts as |G(k)i 〉 → |G(l)

i 〉,whereas H stands for the Hadamard gate. The arrows indi-cates that the procedure may be reiterated conditioned on themeasurement outcomes.

(c) Finally, we perform a measurement on the controlsystem in the computational basis: if the outcome is “1”,the target system is set (up to a global phase) in stateT † |ψ〉L, whereas if the outcome is “G2”, the state of thetarget system is (up to a phase) T |ψ〉L. Analogously tothe implementation of the Hadamard gate, we can keepreiterating the procedure (a)-(c) until we get the desiredgate. The group of transformations that is generated bythe iterative procedure is equivalent (up to global phases)to the cyclic group Z8, i.e. GT ≡

{T j , j = 0, ..., 7

} ∼= Z8.

Notice also that T = ZT 5; thus, if at some step duringthe iteration of the procedure we obtain the unitary T 5,then we can apply the Z-gate (i.e. the classical gate G2),and obtain the desired operation. Taking the latter intoaccount, in the Supplementary Information we prove thatthe probability of generating the required gate T within

m iterations is P = 1−(12

)dm/2e.

The possibility of implementing the H and T gatesimplies the possibility of implementing any single-qubitgate; in order to perform an arbitrary unitary operationon n logical qubits, we still need to show how to imple-ment the two-qubit CNOT-gate.

CNOT-gate. Let us assume that we want to implementthe CNOT gate on the k-th and l-th logical qubits, wherethe latter are in some arbitrary joint state |φ〉L. As weare now dealing with two pairs of target qubits, we willuse explicitly the pair indices k and l to indicate the stateof the control system: for example, if the control system

is prepared in state |G(k)i 〉, then the control and target

interact as:

|G(k)i 〉 ⊗ |φ〉L → |G

(k)i 〉 ⊗ (Gi ⊗ 1) |φ〉L , (16)

whereas, if the control system is in state |G(l)i 〉, the in-

teraction leads to:

|G(l)i 〉 ⊗ |φ〉L → |G

(l)i 〉 ⊗ (1⊗Gi) |φ〉L . (17)

Next, note that the CNOT gate can be imple-mented using the CZ gate as CNOT= (1 ⊗ H)CZ(1 ⊗

H), where CZ= |0〉 〈0| ⊗ 1 + |1〉 〈1| ⊗ Z. Since weknow how to implement the H-gate, it is sufficientto prove the possibility of implementing the CZ-gate.Moreover, the CZ-gate can be decomposed as CZ=12 (1⊗ 1 + 1⊗ Z + Z ⊗ 1− Z ⊗ Z). In order to imple-ment it within our model, we will use the followingscheme, which is pictured in Figure 3.

(a) Let us assume that the joint (control+target)

system is initially in state |G(k)2 〉 ⊗ |φ〉L. We ap-

ply a Hadamard gate on the control system in the{|1〉 , |G(k)

2 〉}

subspace, and we let the control and target

systems interact:

→ 1√2

(|1〉 ⊗ (1⊗ 1) |φ〉L − |G

(k)2 〉 ⊗ (Z ⊗ 1) |φ〉L

).

(18)(b) Once again, we apply the Hadamard gate on the

control system (in the same subspace as before):

1

2(|1〉 ⊗ (1⊗ 1 + Z ⊗ 1) |φ〉L +

|G(k)2 〉 ⊗ (1⊗ 1− Z ⊗ 1) |φ〉L).

(19)

Next, we apply a unitary transformation on the control

system which acts as |G(k)i 〉 → |G

(l)i 〉, in order for it to

implement transformations on the l-th qubit in the sub-sequent interaction. We then let the control and tar-get systems interact and apply a Hadamard gate in the{|1〉 , |G(l)

2 〉}

subspace. The output state is thus:

1

2√

2(|1〉 ⊗ (1⊗ 1 + Z ⊗ 1− 1⊗ Z + Z ⊗ Z) |φ〉L +

|G(l)2 〉 ⊗ (1⊗ 1 + Z ⊗ 1 + 1⊗ Z − Z ⊗ Z) |φ〉L).

(20)

(c) A final measurement of the control yields the tar-

get system either in state CZ |φ〉, or in state CZ|φ〉,where we introduced the unitary operator CZ =12 (1⊗ 1 + Z ⊗ 1− 1⊗ Z + Z ⊗ Z). By reiterating theprocedure, the only new transformations which are gen-

erated are 1 and CZ′≡ CZ × CZ. It is easy to check

that the resulting group GCNOT ≡{

1,CZ, CZ, CZ′}

is

equivalent to P/Z4. The latter implies that the proba-bility of implementing the correct transformation afterm trials is the same as the one needed for implementing

the H gate, i.e. P (m) = 1 −(12

)dm/2e, as proved in the

Supplementary Information.Measurements of the logical qubits – We have shown

that our model enables the implementation of any uni-tary transformation acting on n logical qubits. At theend of the computation, one still needs to be able to readout the state of the qubits, i.e. to perform a projec-tive measurement of the logical qubits in the computa-tional basis {|0〉L , |1〉L}. Due to our simple definition ofthe logical subspace, the required measurement can beimplemented straightforwardly as follows: we perform a

7

(classical) projective measurement on the target qubits inthe classical basis; measurement outcomes |10〉 and |11〉correspond to the logical outcome “0”, whereas outcomes|00〉 and |01〉 correspond to the logical outcome “1” (pro-jective measurements on more logical qubits then followanalogously).

Computational cost – Now we will briefly comment onthe computational cost of our model. Consider that wewant to simulate a quantum circuit of size K, i.e. a se-quence of K gates, each drawn from the standard set{H,T,CNOT}. As proved before, each elementary gatecan be implemented within our model with probability

P (m) = 1 −(12

)dm/2e, where m is the number of iter-

ations of the general procedure (a)-(c). Furthermore,n logical qubits can be initialized within m0 iterations

per qubit with probability(1−

(12

)m0)n

. Therefore, thetotal probability of correctly implementing the desiredcircuit is:

P (m0, ...,mK) =

(1−

(1

2

)m0)n K∏

i=1

(1−

(1

2

)dmi/2e),

(21)where mi is the number of iterations involved in the im-plementation of the i-th gate. Next, we want to estimatethe number of iterations needed in order to achieve prob-ability 1 − δ, for some δ � 1. In order to simplify theanalysis, we will take the number of iterations spent oneach gate to be equal, i.e. mi = mj = m,∀i, j, and wewill set this number to be m = 2m0. Inserting thesespecifications into equation (21) and equating the latterto the desired probability of success we obtain the fol-lowing condition:(

1−(

1

2

)m0)n+K

= 1− δ. (22)

By expanding the latter equation up to first order in δ,we get m0 = log

(K+nδ

). Consequently, the total number

M of gates required to simulate a circuit of size K is:

M = α(2K + n)log

(K + n

δ

), (23)

where the factor α = O(1) takes into account the factthat each iteration involves around 3 to 8 gates appliedto the control and target systems.

PARITY AND SWAP COMPUTING

In the previous section we showed that one can effi-ciently perform universal quantum computation by co-herently controlling (i.e. lifting) the classical set G in-troduced in equation (9). It is interesting to note thatthe operations pertaining to the available set G can beimplemented by the so called parity computer, the latterbeing a classical device that can implement only NOTand CNOT gates (and is capable of computing all parity-preserving affine transformations) [22, 23]. Our result

thus implies that if one is given a parity computer oper-ating on 2n bits, one can achieve full quantum computa-tion by supplementing it with log(2n+ 1) control qubits.This implication draws a parallel between our work andthe study of the “computational power of correlations”[22], as the latter shows that one can achieve universalclassical computation by aiding a parity computer withthree-qubit GHZ-states.

We will now briefly explain how our result implies thateven a computer that is only capable of implementingSWAP gates can be lifted to universal quantum compu-tation2. Namely, it is sufficient to note that the classicalset G can be realized in an alternative manner: instead oftaking the available operations to be single-qubit NOT-gates and two-qubit CNOT-gates, we can use SWAPgates, as follows. We will encode one logical qubit intoa two-dimensional subspace of a Hilbert space of fourqubits, as:

|0〉L ≡1√2

(|1000〉 − |0100〉) ,

|1〉L ≡1√2

(|0010〉 − |0001〉) .(24)

Let us define G1 = SWAP13SWAP24 and G2 =SWAP12, where SWAPij is a SWAP gate acting on thei-th and j-th qubits. It is straightforward to check thatthese gates act on the logical qubit as G1

∼= X andG2∼= −Z. Therefore, one can simply take all the results

from the previous sections and prove the possibility ofperforming universal computation by coherently control-ling SWAP gates (albeit, in this case, in order to performcomputation on n logical qubits, one requires 4n targetqubits). Therefore, if one is given a device that can im-plement only SWAP gates on 4n bits (and is thus capableof generating the trivial class of reversible classical trans-formations according to the classification in [23]), one canlift it to universal quantum computation by aiding it withlog(2n+1) control qubits. We want to point out the sim-ilarity between our conclusions and the ones provided in[34], where the authors presented a deterministic schemethat uses SWAP gates for universal quantum computingwith bosonic systems.

CONCLUSION AND OUTLOOK

Now we will summarize and discuss the results pre-sented in this work. Our goal is to show that one canachieve universal quantum computation on a target sys-tem via the coherent control of classical operations. Westarted by restricting the set of available operations act-ing on the target system to be classical, and we showed

2 A SWAP gate is a two-qubit gate that acts as SWAP|x1x2〉 =|x2x1〉.

8

how a control system can be used to implement trans-formations that lie outside of the available set. We pro-ceeded by showing that a particular set of gates, whichconsists of local NOT and CNOT gates acting on 2n tar-get qubits, can be lifted to the full unitary group actingon n logical qubits. The Hilbert space of the controlsystem is only (2n + 1)-dimensional (i.e. composed oflog(2 + 1) qubits), and the transformations available onthe latter involve only U(2) gates acting in various two-dimensional subspaces. Moreover, all elementary gates(including the initialization of the logical qubits) featurea repeat-until-success strategy, with probabilities of suc-cess that converge exponentially fast to 1 with the num-ber of repetitions: more precisely, in order to simulate acircuit of size K on n qubits with probability 1− δ, oneneeds to act with O((2K + n)log

(K+nδ

)) gates on the

control and target systems. Finally, after noting that theclassical transformations G can be executed by a paritycomputer, we showed that our results imply that evena device that implements only SWAP gates on 4n bitscan be raised to full quantum computing on n qubits, byaiding it with log(2n+ 1) control qubits.

The procedure of enlarging a subset G of available oper-ations to a larger (semi)group G by using a control systemwas named lifting, which may as well be defined for anysubset G of the unitary group, and not only for classicaltransformations. It would thus be interesting to analyzethe lifting procedure in more general terms: one may takean arbitrary subset of the unitary group and inspect the(semi)group that one obtains by using a control system.

Furthermore, one should take into account the efficiencyof the procedure, i.e. the dimension of the control sys-tem and the probability of the successful gates’ imple-mentations. Such an analysis would contribute to theunderstanding and unification of the multifaceted rolesthat the coherent control of operations plays in quantuminformation processing.

Additionally, it would be interesting to inspect poten-tial experimental implementations of our computationalscheme, e.g. in the context of light-assisted [19, 35–39]and matter-assisted quantum computation [40].

ACKNOWLEDGEMENTS

We thank Hoi-Kwan (Kero) Lau for bringing our at-tention to the related work presented in [34]. The au-thors thank Anton Zeilinger, Elizabeth Agudelo, FarajBakhshinezhad and Philip Taranto for valuable dis-cussions. This work was supported by the AustrianAcademy of Sciences (OAW). X.G. acknowledges sup-

port from the Austrian Academy of Sciences (OAW)and the Joint Centre for Extreme Photonics (JCEP).S.H. acknowledges support from the Austrian ScienceFund (FWF) through BeyondC-F7112. B.D. acknowl-edges support from an ESQ Discovery Grant of the Aus-trian Academy of Sciences (OAW) and the Austrian Sci-ence Fund (FWF) through BeyondC-F7112.Note on author contributions. S.H. and X.G. con-

tributed equally to this work.

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10

SUPPLEMENTARY INFORMATION

Probabilities of successfully implementing theelementary gates

In this section we will compute the probability of suc-cessfully implementing each element of the standard setof gates using a finite number of iterations of the proce-dures described in the main text.

H and CNOT gates – As described in the main text,the iteration of the procedures involved in the implemen-tation of the H and CNOT gates lead respectively to thegeneration of groups GH and GCNOT , which are bothequivalent to P/Z4. Here we will focus on the group

GH ={

[1], [H], [H], [Y ]}

; the result for the CNOT-gate

then follows analogously.

Our aim is to answer the following question: whatis the total probability P (m) of implementing the re-quired gate [H], if we have m iterations of the procedureat disposal? Elementary probability theory implies thatP (m) =

∑mi=1 P

(i), where P (i) is the probability of im-plementing the correct transformation after i iterations,given that the transformation has not been implementedin the previous (i−1) iterations. Let us suppose that theinitial state of the target system is |ψ〉; the first iterationof the procedure described in the main text outputs thestate [H] |ψ〉 or [H] |ψ〉, each with probability 1/2: thusP (1) = 1/2. In the case that the outcome of the latter it-

eration is [H], we reiterate the procedure once more andobtain results |ψ〉 or [Y ] |ψ〉, which implies that P (2) = 0.

Another reiteration yields [H] |ψ〉 or [H] |ψ〉, each withprobability 1/2: therefore P (3) = 1/4. It is then easy togeneralize the latter and see that P (i) = 0 for even i, and

P (i) =(12

)di/2efor odd i. The total probability is thus

P (m) =∑dm/2ej=1

(12

)j= 1−

(12

)dm/2e.

T-gate – The procedure involved in the implementationof the T-gate generates the cyclic group

{T j , j = 0, ..., 7

}.

Every iteration of the procedure implements transforma-tions T or T †, each with probability 1/2. The overallprocess can thus be visualized as a random walk of a“particle” on 8 uniformly distributed points on a circle,as in Figure 4. The walk starts at point T 0 and eachiteration generates one step in the clockwise or counter-clockwise direction. Moreover, notice that T = ZT 5:this implies that if the particle finds itself at point T 5,we can easily obtain the desired gate T , as one of theavailable classical gates is isomorphic to the Z gate. Ourgoal is thus to compute the total probability of the par-ticle being at points T 1 or T 5 after m steps of the walk.Similarly as in the previous paragraph, the total proba-bility is P (m) =

∑mi=1 P

(i), where P (i) is the probabilityof the particle being in state T 1 or T 5 after i steps, giventhat it has not passed through these states in the previ-ous (i − 1) steps of the walk. Next, notice that in orderfor the particle to find itself in state T 1 or T 5 at the i-thstep, it must necessarily have been either at T 0 or at T 6

𝑇0

𝑇1𝑇2

𝑇3

𝑇4

𝑇5

𝑇6𝑇7

FIG. 4. Implementation of the T -gate illustrated as a ran-dom walk of a “particle” on eight uniformly distributed pointson a circle. The walk starts at point T 0 and each iterationmoves the particle by one step either in the clockwise or coun-terclockwise direction with uniform probability. The walk isterminated when the particle reaches points T 1 or T 5.

at the (i− 1)-th step. Therefore:

P (i) =1

2

(P

(i−1)0 + P

(i−1)6

), (25)

where P(i−1)j is the probability of the particle being lo-

cated at point T j at the (i− 1)-th step of the walk. Our

goal now is to calculate P(i−1)j , for j = 0, 6.

Let us first define the space of normalized probabilis-tic states of the particle as a convex space St whichcan be embedded in an 8-dimensional vector space V,i.e. St ⊂ V. Next, we introduce an orthonormal basis{~ei, i = 0, 1, ..., 7}, such that, if the particle is in the prob-abilistic state ~p ∈ St, then ~ek · ~p is the probability of itbeing located at point T k. Moreover, the evolution ofthe particle is represented by a sequential application ofa stochastic map M : St→ St.

Now, notice that we are interested exclusively in thewalk of the particle through points T 0, T 7 and T 6; wewill thus focus on the convex subspace Q ⊂ St, whichis spanned by the basis vectors {~e0, ~e7, ~e6}. With all ofthis said, it is easy to see that the evolution operator M ,when restricted to the subspace Q, takes the followingform (written in the basis {~e0, ~e7, ~e6}):

MQ = 12

0 1 01 0 10 1 0

.

Note that the latter operator is not stochastic, asthe walk can map the system outside of the (three-dimensional) domain that we are considering.

Now we are finally ready to compute P(i−1)0 and P

(i−1)6 .

Namely, if the particle is initially in state ~v, then the stateafter m iterations of the walk is (MQ)m~v, given that theparticle has not been mapped outside of our three pointsin the previous steps. Therefore, since the particle is

11

initially in state ~e0 (i.e. at point T 0), it follows that

P(i−1)j = ~ej · (MQ)i−1~e0. Diagonalizing the matrix MQ

and calculating its power yields:

~e0 · (MQ)k~e0 = ~e6 · (MQ)k~e0 =

{0, for odd k;(12

)k/2+1, for even k.

(26)

Together with equation (25), the latter implies that

P (i) = 0, for even i, and P (i) =(12

)di/2e, for odd i.

Finally, we obtain P (m) =∑dm/2ej=1

(12

)j= 1−

(12

)dm/2e.