Asymptotically Improved Grover’s Algorithm in AnyDimensionalQuantum System with Novel Decomposed𝑛-qudit Toffoli Gate
AMIT SAHA, A. K. Choudhury School of Information Technology, University of Calcutta, IndiaRITAJIT MAJUMDAR, Advanced Computing & Microelectronics Unit, Indian Statistical Institute, IndiaDEBASRI SAHA, A. K. Choudhury School of Information Technology, University of Calcutta, IndiaAMLAN CHAKRABARTI, A. K. Choudhury School of Information Technology, University of Calcutta,IndiaSUSMITA SUR-KOLAY, Advanced Computing & Microelectronics Unit, Indian Statistical Institute, India
The progress in building quantum computers to execute quantum algorithms has recently been remarkable.Grover’s search algorithm in a binary quantum system provides considerable speed-up over classical paradigm.Further, Grover’s algorithm can be extended to a 𝑑-ary (qudit) quantum system for utilizing the advantage oflarger state space. In a qudit quantum system, an 𝑛-qudit Toffoli gate plays a significant role in the accurateimplementation of Grover’s algorithm. In this paper, a generalized 𝑛-qudit Toffoli gate has been realized usingqudits to attain a logarithmic depth decomposition without ancilla qudit. The circuit for Grover’s algorithmhas then been designed for any 𝑑-ary quantum system, where 𝑑 ≥ 2, with the proposed 𝑛-qudit Toffoli gate toobtain optimized depth compared to earlier approaches. The technique for decomposing an 𝑛-qudit Toffoligate requires access to higher energy levels, making the design susceptible to leakage error. The performanceof the decomposition for the unitary and erasure models of leakage noise has been studied as well.
Additional Key Words and Phrases: Grover’s Algorithm, Toffoli Decomposition, Qudit System, QuantumLeakage Error
ACM Reference Format:Amit Saha, Ritajit Majumdar, Debasri Saha, Amlan Chakrabarti, and Susmita Sur-Kolay. 2021. AsymptoticallyImproved Grover’s Algorithm in Any Dimensional Quantum System with Novel Decomposed 𝑛-qudit ToffoliGate. 1, 1 (August 2021), 22 pages. https://doi.org/10.1145/nnnnnnn.nnnnnnn
1 INTRODUCTIONThe proliferation of quantum algorithms is gradually grabbing the eye of researchers. Quantumcomputer hardware is now available for physical implementation of these algorithms to attainsignificant speedups [44]. Grover’s search algorithm [24] is a good case in point. Its goal is tosearch data from an unstructured database, and gives quadratic speed-up compared to the classicalcounterparts.
Authors’ addresses: Amit Saha, A. K. Choudhury School of Information Technology, University of Calcutta, Kolkata, India,700106, [email protected]; Ritajit Majumdar, Advanced Computing &Microelectronics Unit, Indian Statistical Institute,Kolkata, India; Debasri Saha, A. K. Choudhury School of Information Technology, University of Calcutta, Kolkata, India,700106; Amlan Chakrabarti, A. K. Choudhury School of Information Technology, University of Calcutta, Kolkata, India,700106; Susmita Sur-Kolay, Advanced Computing & Microelectronics Unit, Indian Statistical Institute, Kolkata, India.
Classical computers are designed on transistors, which deal with binary bits at the physical level.Quantum computers are designed to deal with qubit technology. Albeit, the fundamental physicsbehind the quantum system is not inherently binary, on the contrary, a quantum system can havean infinite arity of discrete energy levels. In reality, the limitation lies in the fact that we need tocontrol the system as per our needs. Including additional discrete energy levels for the purposeof computation helps us to realize the qudit technology quite comprehensively, which makes thesystem more flexible with data storage and faster in processing of quantum information.In this work, we consider the implementation of Grover’s search algorithm generalised to any
dimension. Qudit technology deals with 𝑑-ary quantum systems, where 𝑑 ≥ 2 [7]. For providinga larger state space and simultaneous multiple control operations, we graduate to qudits whicheventually reduce the circuit complexity and enhance the efficiency of quantum algorithms [9, 35, 36,48]. For example, 𝑁 qubits can be expressed as 𝑁
𝑙𝑜𝑔2𝑑qudits, which shaves off by a 𝑙𝑜𝑔2𝑑-factor from
the run-time of a quantum algorithm [37, 38]. The 𝑑-ary quantum computing system can be realizedon various physical technologies, for instance, continuous spin systems [1, 4], superconductingtransmon technology [32], nuclear magnetic resonance [17, 20], photonic systems [19], ion trap[31], topological quantum systems [6, 13, 14] and molecular magnets [34].In this paper, we have designed an efficient quantum circuit for Grover’s algorithm using the
proposed novel decomposition of an 𝑛-qudit Toffoli gate [42]. For physical implementation of𝑛-qudit Toffoli gate, it is of utmost importance to decompose it into one-qudit and/or two-quditgates. In [23], authors have proposed a qubit-qutrit approach to decompose a generalized Toffoligate, which we have extended to 𝑛-qudit Toffoli decomposition with the use of |𝑑⟩ and |𝑑 + 1⟩quantum states as temporary storage. This is a novel approach and achieves optimized depth. Bysimply adding a discrete energy level, we can easily have a higher dimension quantum state fortemporary use, since these are present only as intermediate states in a qudit system. However,the input and output states are qudits, and only in the intermediate operations, we introduce the|𝑑⟩ and |𝑑 + 1⟩ quantum state of 𝑑 + 2-ary quantum system without hampering the operation ofinitialization and measurement on physical devices. By introducing the 𝑑 + 2-ary quantum system,the constraint of arity specific Toffoli decomposition can be avoided. To the best of our knowledge, itis a first of its kind approach. As the 𝑑-ary system may need to occasionally access states beyond the𝑑-ary computational space - an engineering challenge, it makes the system particularly susceptibleto leakage error [51, 54]. We have shown the effect of erasure and unitary leakage model on theproposed implementation of generalised Grover’s search.
Our contributions are the following:
• a novel technique to decompose a generalized 𝑛-qudit Toffoli gate into a logarithmic depthand no-ancilla qudit equivalent circuit — as an example, a 8-qubit Toffoli (𝐶7𝑁𝑂𝑇 ) gaterealization and a comparative study depicts that our approach is better than the existingapproaches in terms of a constant factor of gate cost reduction;
• the design of a circuit for Grover’s search algorithm in any 𝑑-ary quantum system using theproposed decomposed 𝑛-qudit Toffoli gate so that we can reduce the logarithmic factor inthe time complexity of Grover’s algorithm;
• study of the performance of the decomposition technique for the erasure and unitary modelof leakage error, keeping aside the noise mitigation techniques which are not addressed here.
The structure of the paper is as follows. Section 2 describes the universal qudit gates. Section 3defines Grover’s search algorithm in a 𝑑-ary quantum system. Section 4 illustrates the decompo-sition of the proposed 𝑛-qudit Toffoli gate and its comparative analysis. The performance of thedecomposition under leakage noise is presented in Section 5. Section 6 captures our conclusions.
2 GENERALIZED QUDIT GATESA qudit is the unit of quantum information for 𝑑-ary quantum system[33, 52]. Qudit states canbe expressed by a vector in the 𝑑 dimensional Hilbert space H𝑑 [30, 38]. The vector space is thespan of orthonormal basis vectors {|0⟩ , |1⟩ , |2⟩ , . . . |𝑑 − 1⟩}. The general form of qudit state can bedescribed as
where |𝛼0 |2 + |𝛼1 |2 + |𝛼2 |2 + · · · + |𝛼𝑑−1 |2 = 1 and 𝛼0, 𝛼1, . . . , 𝛼𝑑−1 ∈ C𝑑 . An overview of generalizedqudit gates is presented in this section. The generalisation can be defined as discrete quantum statesof any arity [8]. Unitary qudit gates [15, 28] are applied on qudits to modify the quantum statein a quantum algorithm [3]. For logic synthesis of Grover’s algorithm in 𝑑-ary quantum system,one needs to consider one-qudit generalized gates such as NOT gate (𝑋𝑑 ), phase-shift gate (𝑍𝑑 ),Hadamard gate (𝐹𝑑 ), two-qudit generalized CNOT gate (𝐶𝑋,𝑑 ) and Generalized 𝑛-qudit Toffoli gate(𝐶𝑛
𝑋,𝑑). These gates are defined next.
2.1 Generalized NOT Gate𝑋𝑑 is the generalized NOT or increment gate [45], represented by a (𝑑 × 𝑑) matrix is as follows:
ª®®®®®®¬2.3 Generalized Hadamard Gate𝐹𝑑 is the generalized quantum Fourier transform or generalized Hadamard gate [18, 35], whichproduces the superposition of the input basis states. Its (𝑑 × 𝑑) matrix representation is as follows :
ª®®®®®®¬2.4 Generalized CNOT GateQuantum entanglement is a phenomenal property of quantum mechanics, and can be achievedby a controlled NOT (CNOT) gate in a binary quantum system. For 𝑑-ary quantum systems, thebinary 2-qubit CNOT gate is generalised to the 𝐼𝑁𝐶𝑅𝐸𝑀𝐸𝑁𝑇 gate:INCREMENT |𝑥⟩ |𝑦⟩ = |𝑥⟩ |(𝑦 + 1) mod 𝑑⟩, if 𝑥 = 𝑑 − 1, and = |𝑥⟩ |𝑦⟩, otherwise [16].The (𝑑2 × 𝑑2) matrix representation of the generalized CNOT 𝐶𝑋,𝑑 gate is as follows:
ª®®®®®®¬2.5 Generalized 𝑛-qudit Toffoli GateNext, we extend the generalized CNOT or 𝐼𝑁𝐶𝑅𝐸𝑀𝐸𝑁𝑇 further to operate over 𝑛 qudits as ageneralized 𝑛-qudit Toffoli gate 𝐶𝑛
𝑋,𝑑[42]. For 𝐶𝑛
𝑋,𝑑, the target qudit is incremented by 1 (mod 𝑑)
only when all𝑛−1 control qudits have value𝑑−1. The (𝑑𝑛×𝑑𝑛) matrix representation of generalized𝑛-qudit Toffoli gate is as follows:
ª®®®®®®¬Due to technology constraints, a multi-controlled Toffoli gate can be replaced by an equivalent
circuit comprising one-qudit and/or two-qudit gates, albeit at first the multi-controlled Toffoli hasto be decomposed into a set of Toffoli gates for any dimensional quantum system [29]. In Figure 1,we have shown an example of the state-of-the-art approach of decomposition of an 8-qubit Toffoligate with the help of an intermediate qutrit state [2, 22]. The equivalent circuit temporarily storesinformation directly in the qutrit |2⟩ state for the controls.In the schematic diagram of the circuit, we have used a circle to represent the control, and a
rectangle to represent the target. As shown in Figure 1, for a Toffoli gate, we have used ′1′ in the
control circle to represent two qubit controls, and ’+1’ in the target box to represent the incrementoperator.
The decomposed circuit can be treated as a binary tree of gates which establishes the logarithmicdepth for a multi-controlled Toffoli gate. It has the property that the intermediate qubit of eachsub-tree as well as the root can only be raised to |2⟩ if all of its seven control leaves are |1⟩. Inorder to verify this property, we perceive that the qubit 𝑞4 can only become |2⟩ if and only if it wasoriginally |1⟩, and 𝑞2 and 𝑞6 qubits were previously |2⟩. Then at the subsequent level of the tree, weobserve that (i) qubit 𝑞2 could have been |2⟩ only if it was previously |1⟩, and both 𝑞1 and 𝑞3 were|1⟩ earlier, (ii) qubit 𝑞6 could have been |2⟩ only if it was previously |1⟩ and both 𝑞5 and 𝑞7 qubitswere |1⟩ earlier. If any of the controls were not |1⟩, the |2⟩ state would fail to move to the root ofthe tree. Hence, the𝐶𝑁𝑂𝑇 gate toggles the target qubit only if all controls are |1⟩. The right half ofthe circuit is the mirror circuit to restore the control qubits to their original states. The authorsin [16] have further decomposed their ternary Toffoli gate into 13 1-qutrit and 2-qutrit gates forphysical implementation.
Fig. 1. Decomposition of 8-qubit Toffoli [22]
These works are nonetheless restricted to binary quantum systems as they have mentioned theuse of qutrits only. In our proposed approach, we have generalized the decomposition for anydimensional quantum system.
3 GENERALIZED GROVER’S ALGORITHM IN 𝑑-ARY QUANTUM SYSTEMWe present the details of our proposed generalized Grover’s algorithm in 𝑑-ary quantum systemhere. The algorithm has two sub-parts: Oracle and diffusion [24]. Formally, Grover’s algorithm forsearching in an unstructured database can be defined as follows: given a collection of unstructureddatabase elements 𝑥 = 1, 2, . . . , 𝑁 , and an Oracle function 𝑓 (𝑥) that acts on a marked element 𝑠 asfollows [27],
𝑓 (𝑥) ={1, 𝑥 = 𝑠,
0, 𝑥 ≠ 𝑠,(2)
find the marked element with as few calls to 𝑓 (𝑥) as possible [24, 27]. The database is encoded intoa superposition of quantum states with each element being assigned to a corresponding basis state.Grover’s algorithm searches over each possible outcome, which is represented as a basis vector |𝑥⟩in an 𝑛-dimensional Hilbert space in 𝑑-ary quantum system. Correspondingly, the marked elementis encoded as |𝑠⟩. Thus, after applying unitary operations as an oracle function to the superpositionof the different possible outcomes, the search can be done in parallel. Then the generalized diffusion
operator, which is also known as inversion about the average operator, amplifies the amplitudeof the marked state to increase its measurement probability using constructive interference, withsimultaneous enfeeblement of all other amplitudes, and searches the marked element in 𝑂 (
√𝑁 )
steps, where 𝑁 = 𝑑𝑛 [27].The circuit diagram for the generalized Grover’s algorithm in a𝑑-ary quantum system is presented
in Figure 2, where at least 𝑛 + 1 qudits are required. More elaborately, the steps of the Grover’salgorithm are as follows:
Fig. 2. Generalized Circuit for Grover’s algorithm in 𝑑-ary quantum system
Initialization: The algorithm starts with the uniform superposition of all the basis states onthe 𝑛 input qudits in |0⟩ by incorporating generalized Hadamard or quantum DFT gate. The lastancilla qudit is used as an output qudit which is initialized to 𝐹𝑑 |𝑑 − 1⟩. Thus, we obtain the 𝑑-aryquantum state |𝑎⟩:
|𝑎⟩ = 𝐹 ⊗𝑛𝑑
|0𝑑⟩ =1
√𝑑𝑛
𝑑𝑛∑︁𝑥=1
|𝑥⟩
Oracle query: The oracle (𝑈𝑓 ) of Grover search marks the marked state |𝑠⟩ while keeping allthe other states unaltered, and can be expressed as:
|𝑥⟩𝑈𝑓−−→ (−1) 𝑓 (𝑥) |𝑥⟩
The oracle block𝑈𝑓 as shown in Figure 2 depends on the problem instance. One needs to designthe oracle using Unitary transformation as per requirement.
Diffusion: The diffusion operator of Grover’s search is generic and not problem specific. Asshown in Figure 2, the diffusion operator is initialized with generalized Hadamard (𝐹 ⊗𝑛
𝑑) followed
by 2 |0𝑛⟩ ⟨0𝑛 | − 𝐼𝑑𝑛 and generalized Hadamard (𝐹 ⊗𝑛𝑑
) again. The diffusion operator (𝐷) can beexpressed as:
𝐷 = 𝐹 ⊗𝑛𝑑
[2 |0𝑛⟩ ⟨0𝑛 | − 𝐼𝑑𝑛 ]𝐹 ⊗𝑛𝑑
The matrix representation of generalized diffusion operator [27] for 𝑑-ary quantum system isshown below:
We need to iterate the Grover’s operator 𝑂 (√𝑁 ) times to get the coefficient of the marked state
|𝑠⟩ large enough that it can be obtained from measurement with probability close to 1 and thusconclude the Grover’s algorithm.The generalized circuit for Grover’s diffusion operator in 𝑑-ary quantum system as shown in
Figure 3 can be constructed using generalized Hadamard gate, generalized NOT gate and generalized𝑛-qudit Toffoli gate. As discussed in Section 2, for implementing Grover’s algorithm in technologyspecific physical devices, the 𝑛-qudit Toffoli gate needs to be decomposed using one-qudit or two-qudit gates. While decomposing the 𝑛-qudit Toffoli gate, if the depth and the ancilla qudits increasearbitrarily then the time complexity of Grover’s algorithm also increases, which is undesirable.In the next subsection, we have shown a novel approach for the decomposition of an 𝑛-quditToffoli gate with optimized depth as compared to the state-of-the-art. Thus, the time complexity ofGrover’s algorithm will also be optimized.
Fig. 3. Generalized Circuit for Grover’s Diffusion Operator in 𝑑-ary quantum system [27]
4 IMPROVED CIRCUIT DESIGN FOR GROVER’S SEARCH IN 𝑑-ARY QUANTUMIn order to execute Grover’s algorithm on physical quantum devices, it has to be ideally decomposedusing single-qudit and/or two- qudit gates. It is important to carry out the effective low depth andlow gate count decomposition in near term quantum devices and beyond [46].
4.1 Proposed 𝑛-qudit Toffoli Gate DecompositionThe most important aspect of our proposed work is the decomposition of 𝑛-qudit 𝑑-dimensionalToffoli gate. In the decomposition of generalized Toffoli gate, all the figures below have inputs andoutputs as 𝑑-dimensional qudits, but the states |𝑑⟩ and |𝑑 + 1⟩ may be used in intermediate levelsduring the computation. The idea of keeping 𝑑-ary input/output enables these circuit constructionsto be applied for any already existing 𝑑-ary qudit-only circuits.A generalized Toffoli decomposition in 𝑑-ary system using |𝑑⟩ state is shown in Figure 4. A
similar construction for the Toffoli gate in binary using qutrit is evident from previous state-of-the-art work [22]; we have extended it for 𝑑-ary quantum system. The aim is to carry out an 𝑋𝑑
operation on the target qudit (third qudit) as long as the two control qudits, are both |𝑑 − 1⟩. First, a|𝑑 − 1⟩-controlled 𝑋 +1
𝑑+1, where +1 and 𝑑 + 1 are used to denote that the target qudit is incrementedby 1 (mod 𝑑 + 1), is performed on the first and the second qudits. This upgrades the second qudit to|𝑑⟩ if and only if the first and the second qudits were both |𝑑 − 1⟩. Then, a |𝑑⟩-controlled 𝑋𝑑 gate isapplied to the target qudit. Therefore, 𝑋𝑑 is executed only when both the first and the second quditswere |𝑑 − 1⟩, as expected. The controls are reinstated to their original states by a |𝑑 − 1⟩-controlled𝑋−1𝑑+1 gate, which reverses the effect of the first gate. That the |𝑑⟩ state from 𝑑 + 1-ary quantum
system can be used instead of ancilla to store temporary information, which is the most importantaspect in this decomposition.As in [22], the circuit decomposition of generalized Toffoli gate is realized in terms of ternary
Toffoli gate instead of 1-qutrit and 2-qutrit gates in order to obtain lower circuit depth. But during
Fig. 4. Generalized Toffoli in 𝑑-ary quantum system
simulation, they decomposed the ternary Toffoli gate into six 2-qutrit and seven 1-qutrit physicallyimplementable quantum gates. We have also followed a similar approach for extending the decom-position of generalized 𝑛-qudit Toffoli gate in terms of 𝑑 + 1-ary Toffoli gate. But, the approachof further decomposition of the Toffoli for simulation purpose has not been adopted. Instead, the𝑑 + 1-ary Toffoli gate has been decomposed into 𝑑 + 2-ary CNOT gates. Let us consider a generalizedCNOT gate for 𝑑 + 2-ary quantum system as𝐶+1
𝑋,𝑑+2, where +1 and 𝑑 + 2 denote that the target quditis incremented by 1 (mod 𝑑 + 2) only when the control qudit value is 𝑑 + 1. The ((𝑑 + 2)2 × (𝑑 + 2)2)matrix representation of the 𝐶+1
ª®®®®®®¬For example, let there be a 8-qudit Toffoli gate as shown in Figure 5(a). First, we decompose it as
in [22] as shown in Figure 5(b). Further, we decompose all the 𝑑 + 1-ary Toffoli gate into (𝑑 + 2)-aryCNOT gates as shown in Figure 5(c) with the help of the proposed decomposition of generalizedToffoli in any dimensional quantum system. As shown in Figure 5(c), all the 𝑑 − 1-controlledToffoli gates are decomposed into 𝑑 − 1-controlled and 𝑑-controlled CNOT gates. Similarly, allthe 𝑑-controlled Toffoli gates are decomposed into 𝑑-controlled and 𝑑 + 1-controlled CNOT gates.Thus, with the help of |𝑑⟩ and |𝑑 + 1⟩ quantum state of (𝑑 + 2)-ary system, 𝑋𝑑 is executed if all thecontrolled qudits are in |𝑑 − 1⟩ state. In this manner, an 𝑛-qudit Toffoli gate can be decomposed.Further, the optimized 𝑛-qudit Toffoli gate decomposition has been portrayed in Figure 5(d), wherealongside generalized CNOT gates on same qudits are removed by applying the optimization ruleas described in [43]. Now, if we want to apply our approach to a binary quantum system, thenit could be easily carried out if ququads of quaternary [41] or a 4-ary quantum system comesinto play. Moreover, we have achieved logarithmic depth as well as reduced the constant factorfrom 13 to 2, which is thoroughly discussed in the next subsection with the help of an example.By simulation, we have verified our circuits. The simulation results for the 8-qubit Toffoli gate ofFigure 6(c), appears in the Appendix. In Table 3 of Appendix, we have shown the input and outputstates as well as intermediate states for each time cycle of the circuit for all possible combination
of input states |00000000⟩, |00000010⟩, |00000100⟩, . . . , |11111110⟩. We have shown that only forthe input state |11111110⟩, the output state changes to |11111111⟩, otherwise there is no change ofoutput states for corresponding input states.
4.2 Comparative AnalysisA comparative study of our Toffoli decomposition with some previous works [3, 21, 22, 25, 33, 47, 53]is shown in Table 1. Our work outperforms all of them in terms of depth of the circuit, even the besttill now [22]. We simulate our work taking the conventional construction proposed by Gokhale et al.[23] into account, since it is the benchmark in the ancilla-free frontier zone. The technique makesthe decomposition typically exorbitant in gate count and depth as a large number for constantfactor of gate-count is required as compared to our approach. This is better explained with thefollowing example.
As shown in Figure 6(a), a multi-controlled Toffoli gate with 7 controls and 1 target is considered.Figure 6(b) depicts the decomposition of the generalized 8-qubit Toffoli gate as shown in Figure6(a) with the help of the design proposed by Gokhale et al. [22]. Their circuit temporarily storesinformation directly in the qutrit |2⟩ state of the controls, so does our approach. However, instead
of storing temporary results further with ququads |3⟩ state, they simply decompose their ternaryToffoli into 13 1-qutrit and 2-qutrit gates [16] as mentioned in their paper [22], but in our case, wefurther decompose ternary Toffoli into three 4-ary CNOT gates using ququads |3⟩ state as control,which is shown in Figure 6(c). As shown in 6(d), further optimization can lead to the minimizationof the constant factor of gate-count from 13 to 2 for single Toffoli decomposition and our approachbecomes qudit generalizable.Our circuit construction as shown in 6(c) or 6(d), as in [22], can also be interpreted as a binary
tree of gates. More elaborately, the inputs/outputs are qubits, but we grant inhibition of the |2⟩ and|3⟩ ququads states in between. The circuit maintains a tree structure and has the property that theintermediate qubit, of each sub-tree as well as root can only be raised to |2⟩ if all of its seven controlleaves were |1⟩. In order to verify this property, we perceive that the qubit 𝑞4 can only become |2⟩if and only if it was originally |1⟩ and qubit 𝑞6 was previously |3⟩. At the following level of the tree,we see qubit 𝑞6 could have only been |3⟩ if it was previously |1⟩ and both 𝑞3 and 𝑞7 qubits were |2⟩before. If any of the controls were not |1⟩, the |2⟩ or |3⟩ states would fail to move to the root ofthe tree. Hence, the 𝑋 gate is only carried out if all controls are |1⟩. The right half of the circuitundergoes computation to get back the controls to their original state. The construction appliesmore generally to any multi-controlled𝑈 gate.
After each succeeding level of the tree structure, the number of qubits under inspection is reducedby a factor of ∼ 2. This leads to the circuit depth being logarithmic in 𝑛, where 𝑛 is the number ofcontrols. On top of that, each ququad is operated on by a small constant number of three gates, sothe total number of gates is optimized. Wang et al. [53] has also mentioned about 𝑛-qudit Toffolidecomposition before us. In Table 1, we have shown that our approach gives better result in termsof depth and gate cost than theirs. Further, we have shown a 8-qudit Toffoli decomposition but thiscan be extended to 𝑛-qudit also.The proposed 𝑛-qudit Toffoli decomposition is novel not only for its logarithmic depth opti-
mization as compared to [53], but also the maximum number of CNOT gates required is 2 ∗ 𝑛 − 3,which is less compared to 2 ∗ 𝑛 + 1 needed by the decomposition by Wang et al. [53]. We have alsoillustrated examples with 16-qudit Toffoli and 32-qudit Toffoli decomposition elaborately in Figures7 and 8 respectively. Mapping the structure to a binary tree topology helps in establishing the claimfor logarithmic depth.
Table 1. Asymptotic comparison of 𝑛-controlled gate decomposition.
This Work Gokhale[22] Gidney [21] He [25] Barenco [3] Wang [53] Lanyon [33], Ralph [47]Depth log2 𝑛 log2 𝑛 𝑛 log2 𝑛 𝑛2 𝑛 𝑛
Ancilla 0 0 0 𝑛 0 0 0Qudit Types Controls are qudits Controls are qutrits Qubits Qubits Qubits Controls are qutrits/qudits Target is 𝑑 = 𝑛-level quditConstants 2 13 9 9 9 2 9
Recall that in a 𝑑-ary quantum system, generalized Grover’s algorithm for search over 𝑁 un-structured database items requires 𝑂 (
√𝑁 ) iterations of Grover’s operator, where 𝑁 = 𝑑𝑛 and
𝑑 ≥ 2. As discussed earlier, the Grover’s operator is the combination of the oracle and the diffusion.However in each iteration, Grover search has multi-controlled Toffoli gate in diffusion operatorwith𝑀 = ⌈log𝑑 𝑁 ⌉ controls [27]. In other words, each of the iterations has 𝑛-qudit Toffoli gate or(log𝑑 𝑁 )-qudits in Grover’s diffusion operator as already discussed in Figure 3. The best knownToffoli decomposition in qudit system [53], specifically in ternary quantum system shows that thedepth of the realized circuit is linear, i.e., log𝑑 𝑁 or 𝑛. But, our decomposition of the 𝑛-qudit Toffoligate leads to a reduction of the log𝑑 𝑁 factor in Grover’s algorithm to a log2 log𝑑 𝑁 factor in eachiteration. Hence, our proposed 𝑛-qudit Toffoli decomposition leads to a reduction by a logarithmic
factor, i.e., 𝑂 (log2 log𝑑 𝑁 ) in the time complexity of generalised Grover search, compared to theprevious works [26, 27], as shown in Table 2.
Table 2. Comparison of circuit depth, i.e., worst case time complexity for 𝑑-ary Grover’s search, 𝑑 > 2.
This Work Hunt [26] Ivanov [27]Depth log2 (log𝑑 𝑁 ) log𝑑 𝑁 log𝑑 𝑁
We have discussed the proposed decomposition of the 𝑑-ary Toffoli gate, and showed that it issuperior to other decomposition in the literature in terms of the depth as well as the number ofancilla qudits required. We have considered every gate to be ideal, and therefore, the success rateof the Grover’s algorithm remains same as for any other decomposition of the Toffoli Gate. It hasbeen shown in the literature that if the input state is an entangled state (e.g. GHZ or W state), thenthe success rate of the algorithm is (1 − 1
𝑛)𝑛−1 where 𝑛 is the number of qudits [10–12]. However,
unlike [10–12], in this paper, we start the algorithm with the equal superposition of all the basisstates, and hence the success probability is ∼ 1 after
that the action of leakage error can lower the success rate of this algorithm. For Unitary modelof the leakage error, we show that the success rate of the algorithm is lowered to some extent,whereas for the Erasure model the success rate drops exponentially with the number of qudits.
5 ACTION OF LEAKAGE ERROR ON PROPOSED DECOMPOSITION OF 𝑛-QUDITTOFFOLI GATE
Any quantum system is susceptible to different types of errors such as decoherence, noisy gates.For a 𝑑-dimensional quantum system, the gate error scales as 𝑑2 and 𝑑4 for single and two qubitgates respectively [22]. Furthermore, for qubits, the amplitude damping error decays the state |1⟩ to|0⟩ with probability _1. For a 𝑑-dimensional system, every state in level |𝑖⟩ ≠ |0⟩ has a probability_𝑖 of decaying. In other words, the usage of higher dimensional states penalizes the system withmore errors. Nevertheless, the effect of these errors on the used decomposition of Toffoli gate hasbeen studied by Gokhale et al. [22]. They have shown that although the usage of qudits lead toincreased error, the overall error probability of the decomposition is lower than the existing onessince the number of ancilla qudits and the depth are both reduced.Gokhale et al. [22] have considered typical errors but not the leakage error. A qudit is the
span of {|0⟩ , |1⟩ , . . . , |𝑑 − 1⟩}, called the computational subspace. However, it is an engineeringchallenge to prepare such a computational subspace. In general, the prepared system is a muchlarger space of dimension D such that D = 𝑑 ⊕ 𝑑𝑙 where ⊕ denotes the direct sum. The qudit isembedded in the much larger space, and the excess 𝑑𝑙 -dimensional subspace, which is the spanof {|𝑑⟩ , |𝑑 + 1⟩ , . . . , |𝐷 − 1⟩}, is termed as the leakage subspace [51, 54]. The system tends to leakfrom the computational subspace to the leakage subspace, i.e. access the higher dimensions, leadingto erroneous results. Such leakage is a serious obstacle for reliable computation since normalprotection schemes against decoherence [39, 40, 49, 50] is unable to correct such a leakage error[51].
In this decomposition, the generalized qudit Toffoli requires access to the 𝑑 + 2-th dimension. Thesystem must be intentionally embedded in a subspace larger than its computational subspace, andthe higher dimension is accessed occasionally. The risk of leakage is high for such a decomposition.We show, however, in the presence of different leakage error models, the proposed decompositioncan lead to completely or partially erroneous result.LetH𝑑 andH𝑑𝑙 be the Hilbert Space associated with the computational and leakage subspace
respectively. The probability of a quantum state 𝜌 to leak out of the computational subspace, alsocalled the leakage rate, under some evolution E is given by [51]
L(E(𝜌))) = 𝑇𝑟 {𝑃𝐻𝑑𝑙E(𝜌)}
where 𝑃𝐻𝑑𝑙is a projector on the leakage subspace. In other words, the above expression gives the
probability of finding the qudit 𝜌 in the leakage subspace after the evolution via a CPTP map E.If the system has not leaked to the leakage subspace, then the expression 𝑇𝑟 {𝑃𝐻𝑑𝑙
E(𝜌)} wouldevaluate to 0 due to the projective measurement.Two typical models of leakage error considers the two scenarios where the system either com-
pletely or partially leaks to the leakage subspace [54]. They are mathematically expressed in termsof erasure model and a unitary model respectively. For the sake of simplicity, we shall henceforthconsider that the leakage subspace is the span of {|𝑑⟩ , |𝑑 + 1⟩}, where 𝑑 is the dimension of thequdit. However, as described henceforth, the dimension of the computation or leakage subspacedoes not contribute to the effect of these two leakage models. The system, at occasion, is raised tothe energy level ≥ 𝑑 for the working principle of the Toffoli gate. Nevertheless, there is a non-zeroprobability of the system to leak into those higher dimensions spontaneously.
5.1 Unitary model of the leakage errorThe model considers a scenario where only the state |𝑑 − 1⟩ can leak into the immediately higherdimension |𝑑⟩, and the state from leakage subspace can return to the state |𝑑 − 1⟩. Assuming equalprobability of both the processes, the corresponding Hamiltonian is
𝐻 =12 ( |𝑑⟩ ⟨𝑑 − 1| + |𝑑 − 1⟩ ⟨𝑑 |)
The unitary corresponding to the Hamiltonian is [54]𝑈 = 𝑒𝑥𝑝 (−𝑖𝐻𝑡) = I − (𝑐𝑜𝑠 (𝑡/2) − 1) ( |𝑑 − 1⟩ ⟨𝑑 − 1| + |𝑑⟩ ⟨𝑑 |)
+𝑠𝑖𝑛(𝑡/2) ( |𝑑⟩ ⟨𝑑 − 1| + |𝑑 − 1⟩ ⟨𝑑 |)where 𝑡 is the time duration for which the Hamiltonian is applied. The corresponding Leakage
Rate of a quantum state 𝜌 , as shown in [54], isL(𝜌 (𝑡)) = 𝑠𝑖𝑛2 (𝑡/2) ⟨𝑑 − 1|𝜌 |𝑑 − 1⟩ . (3)
Let us assume a quantum state𝜓 =∑𝑑−2
𝑖=0 𝛼𝑖 |𝑖⟩ + 𝛼𝑑−1 |𝑑 − 1⟩, where 𝛼𝑖 ∈ C and∑𝑑−1
𝑖=0 |𝛼𝑖 |2 = 1. If𝑝𝑙 is the leakage probability, then the state changes to
𝑑−2∑︁𝑖=0
𝛼𝑖 |𝑖⟩ +√𝑝𝑙 .𝛼𝑑−1 |𝑑 − 1⟩ +
√︁1 − 𝑝𝑙 .𝛼𝑑−1 |𝑑⟩
.The leakage rate, according to Equation 3, is L(𝜌 (𝑡)) = 𝑠𝑖𝑛2 (𝑡/2).𝑝𝑙 .|𝛼𝑑−1 |2. For Grover’s search
algorithm, initially the state is prepared in equal superposition of the basis states. Assuming𝛼𝑑−1 =
1√2 , we have the leakage rate as
L(𝜌 (𝑡)) = 𝑠𝑖𝑛2 (𝑡/2) 𝑝𝑙2.According to our proposed decomposition technique, each 3-qudit Toffoli is designed using 3
CNOT gates. In current IBM superconductor devices, the average time duration of a single CNOTgate is ∼ 930 ns (according to the calibration details of the Melbourne Device). Due to the smalltime duration of each gate, the probability leakage error in the entire circuit of an 𝑛-qudit Grover’sSearch algorithm (for 𝑛 ≤ 50) is negligible. However, a unitary model of leakage error occurs. Weconsider the action of such an error on a 3-qudit Toffoli, which forms the basis of the 𝑛-qudit Toffolirealization. A general 3-qudit Toffoli gate has its input of the form
∑𝑑−1𝑖, 𝑗=0 𝛼𝑖, 𝑗 |𝑖, 𝑗⟩ |𝑡⟩ where the first
two are the control qudits and 𝑡 is the target qudit. The Toffoli gate will change the target onlywhen both the inputs are 𝑑 − 1. Here, we show the action of a Toffoli gate on such a superposition.∑︁
In case there is a leakage error, the probability amplitude of the state |𝑑 − 1, 𝑑 − 1⟩ changes from𝛼𝑑−1,𝑑−1 to 𝑝2 .𝛼𝑑−1,𝑑−1 where 𝑝 depends on whether leakage error occurred on single or multiplequdits. The probability associated with the correct Toffoli action decreases 𝑝 times. Furthermore,
some of the states of the form |𝑙, 𝑑 − 1⟩ |𝑡⟩, where the first qudit can be in any state 0, 1, . . . 𝑑 − 2(the scenario of |𝑑 − 1, 𝑑 − 1⟩ is already discussed), changes to |𝑙, 𝑑⟩ |𝑡⟩. The action of the Toffoligate will lead the target qubit to change from |𝑡⟩ to |𝑡 + 1⟩. Similarly, there can be scenario, wheresome of the states are of the form |𝑑 − 2, 𝑑 − 1⟩ |𝑡⟩. If leakage error can change the first controlqudit to the state |𝑑 − 1⟩. In such a scenario, the action of the Toffoli gate will change the targetqubit from |𝑡⟩ to |𝑡 + 1⟩.In other words, the unitary model of leakage error reduces the probability associated with the
correct action of the Toffoli gate, and associates some probability of incorrect Toffoli gate action.However, due to its dependency on the time of action, the action of the leakage error model seemsto be less severe than the erasure model.
5.2 Erasure model of the leakage errorIn this model, the system leaks completely into the leakage subspace with probability 𝑝𝑙 . If 𝜌 =∑𝑑−1
𝑖=0 𝛼𝑖 |𝜓𝑖⟩ ⟨𝜓𝑖 | and 𝜌𝑙 = |𝜓𝐷⟩ ⟨𝜓𝐷 | be the states of the original system and the system in theleakage subspace respectively, then under the action of an evolution E on the quantum state
E(𝜌) = (1 − 𝑝𝑙 )𝜌 + 𝑝𝑙 |𝜓𝐷⟩ ⟨𝜓𝐷 | (4)In Equation 5, |𝜓𝐷⟩ ∈ span of {|𝑑⟩ , |𝑑 + 1⟩}. The depth of the circuit of an 𝑛-qudit Toffoli
decomposition is 2⌈𝑙𝑜𝑔(𝑛)⌉, where the first half is for the action of the Toffoli gate, and the secondhalf is to restore the system to its original configuration. The probability that the system stays in itscomputation subspace after the action of the Toffoli gate is (1 − 𝑝𝑙 )2 ⌈𝑙𝑜𝑔 (𝑛) ⌉ . Interestingly the decayin probability is not exponential with the number of qudits (due to the ⌈𝑙𝑜𝑔(𝑛)⌉ power). However,the decay can still be substantial for large 𝑛.
In our Toffoli decomposition (Figure 4), a particular qudit𝑞𝑐 is raised to the energy level𝑑 (𝑜𝑟 𝑑+1)if the two corresponding control qudits 𝑞1𝑐 and 𝑞2𝑐 are both in an energy level 𝑑 − 1(𝑜𝑟 𝑑). Then thecorresponding target qudit 𝑞𝑡 undergoes addition by 1 (modulo d) if 𝑞𝑐 = |𝑑⟩ . However, under theerasure model of leakage, it is possible that the qudit 𝑞𝑐 creeps to the energy level 𝑑 or 𝑑 + 1 due tothe leakage error. In such a scenario(i) If the control qudits 𝑞1𝑐 and 𝑞2𝑐 are not in an energy level 𝑑 − 1, even then 𝑞𝑐 is in state 𝑑 (or
𝑑 + 1), and the addition of +1 (modulo d) occurs on the qudit 𝑞𝑡 .(ii) If the control qudits 𝑞1𝑐 and 𝑞2𝑐 are in an energy level 𝑑 − 1, then 𝑞𝑐 , which was already in
energy level 𝑑 due to leakage error, is changed to the energy level 0 by the actions of 𝑞1𝑐 and𝑞2𝑐 . The action on the target qudit 𝑞𝑡 does not take place at all.
Therefore, a leakage error, which is of the form of an erasure model, leads to a completelyerroneous outcome of the Toffoli gate. When the decomposition technique of Toffoli gate is appliedon an 𝑛-qudit Grover’s Search algorithm, there are O(
√𝑁 ) (where 𝑁 = 𝑑𝑛) iterations of the
algorithm and the Toffoli gate is applied in each iteration. The probability of no leakage error inthe entire algorithm procedure is
(1 − 𝑝𝑙 )2 ⌈𝑙𝑜𝑔 (𝑛) ⌉ .√𝑁 . (5)
In Figure 9 and 10, we show the effect of the erasure model of leakage error when the probabilityof error are 𝑝𝑙 = 0.0001 and 𝑝𝑙 = 0.001 respectively.
We observe that this model of leakage error does have an adverse effect on the success probabilityof the computation. However, for this decomposition, the depth of each Toffoli gate is O(𝑙𝑜𝑔𝑛). Inother decomposition, where the depth of each Toffoli gate is O(𝑛) [21] or O(𝑛2) [3], Equation 5changes as (1−𝑝𝑙 )O(𝑛) .
√𝑁 , or (1−𝑝𝑙 )O(𝑛2) .
√𝑁 . In both of these cases, the performance will degrade
even more drastically with the number of qudits. Our O(𝑙𝑜𝑔𝑛) decomposition provides somemitigation to the performance degradation. Nevertheless, the performance degradation is still
Fig. 9. Performance of n-qudit Grover’s Search under erasure model of Leakage error (𝑝𝑙 = 0.0001)
significant, and the future scope to such decomposition, assisted by the usage of higher dimensions,will be to look into efficient protection schemes against such leakage error.
6 CONCLUSIONIn this work, we have proposed a novel approach to decompose a generalized 𝑛-qudit Toffoli gateinto 2-qudit gates with logarithmic depth without using any ancilla qudit. We have shown aninstance of 8-qudit Toffoli gate decomposition to establish the logarithmic depth as an example.We have given a comparative study to establish that our approach is better than the existingstate-of-the-art ones. We have also shown that Grover’s algorithm can be implemented in any 𝑑-aryquantum system with the proposed 𝑛-qudit Toffoli gate to get the advantage of optimized depthas compared to earlier approaches. Using our novel proposed decomposition of 𝑛-qudit Toffoligate, any quantum algorithm can be optimized that employs generalized Toffoli gate. Finally, wehave studied the effect of leakage error on this decomposition technique. Our study shows thatthe decomposition is more fallible to the erasure model of leakage noise than the unitary model.Nevertheless, for low error probability, the gate can operate with high fidelity.
ACKNOWLEDGMENTSThe first author acknowledges the support by the Grant No. 09/028(0987)/2016-EMR-I from CSIR,Govt. of India.
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A SIMULATION RESULTSThe simulation result for 8-qubit Toffoli gate, Figure 6(c), is shown in this section. In Table 3, Wehave shown that only for the input state |11111110⟩, the output state changes to |11111111⟩, whichis highlighted in Table 3, otherwise there is no change of output states for corresponding inputstates. Similarly if we initialize 𝑞8 of Figure 6(c) with |1⟩, then the output state changes to |11111110⟩for the input state |11111111⟩. There is no change of output states for other corresponding inputstates as well. The simulation is carried out on Google Colab platform [5] and the code is availableat https://github.com/N-Qudit-Toffoli-Decomposition.
