# Qudits and High-Dimensional Quantum Computing

^{1}Department of Chemistry, Purdue University, West Lafayette, IN, United States^{2}Department of Physics and Purdue Quantum Science and Engineering Institute, Purdue University, West Lafayette, IN, United States^{3}Institute for Quantum Science and Technology, University of Calgary, Calgary, AB, Canada

Qudit is a multi-level computational unit alternative to the conventional 2-level qubit. Compared to qubit, qudit provides a larger state space to store and process information, and thus can provide reduction of the circuit complexity, simplification of the experimental setup and enhancement of the algorithm efficiency. This review provides an overview of qudit-based quantum computing covering a variety of topics ranging from circuit building, algorithm design, to experimental methods. We first discuss the qudit gate universality and a variety of qudit gates including the pi/8 gate, the SWAP gate, and the multi-level controlled-gate. We then present the qudit version of several representative quantum algorithms including the Deutsch-Jozsa algorithm, the quantum Fourier transform, and the phase estimation algorithm. Finally we discuss various physical realizations for qudit computation such as the photonic platform, iron trap, and nuclear magnetic resonance.

## Introduction

Qudit technology, with a qudit being a quantum version of *d*-ary digits for

Although the qudit system’s advantages in various applications and potentials for future development are substantial, this system receives less attention than the conventional qubit-based quantum computing, and a comprehensive review of the qudit-based models and technologies is needed. This review article provides an overview of qudit-based quantum computing covering a variety of topics ranging from circuit building [39, 61, 71, 89, 133]; algorithm designs [2, 17, 26, 62, 79, 119, 121]; to experimental methods [2, 11, 48, 60, 62, 91, 99, 106]. In this article, high-dimensional generalizations of many widely used quantum gates are presented and the universality of the qudit gates is shown. Qudit versions of three major classes of quantum algorithms—algorithms for the oracles decision problems (e.g., the Deutsch-Jozsa algorithm [121], algorithms for the hidden non-abelian subgroup problems (e.g., the phase-estimation algorithms (PEAs) [26] and the quantum search algorithm (e.g., Grover’s algorithm [79]—are discussed and the comparison of the qudit designs vs. the qubit designs is analyzed. Finally, we introduce various physical platforms that can implement qudit computation and compare their performances with their qubit counterparts.

Our article is organized as follows. Definitions and properties of a qudit and related qudit gates are given in Section 2. The generalization of the universal gate set to qudit systems and several proposed sets are provided in Section 2.1. Then Section 2.2 lists various examples of qudit gates and discusses the difference and possible improvement of these gates over their qubit counterparts. A discussion of the gate efficiency of synthesizing an arbitrary unitary *U* using geometric method is given in Section 2.3. The next section, Section 3, provides an introduction to qudit algorithms: a single-qudit algorithm that finds the parity of a permutation in Section 3.1.1, the Deutsch-Josza algorithm in Section 3.1.2, the Bernstein-Vazirani algorithm in Section 3.1.3, the quantum Fourier transform in Section 3.2.1, the PEA in Section 3.2.2 and the quantum search algorithm in Section 3.3. Section 4 is a section focused on the qudit quantum computing models other than the circuit model, which includes the measurement-based model in Section 4.1, the adiabatic quantum computing in Section 4.2 and the topological quantum computing in Section 4.3. In Section 5, we provide various realizations of the qudit algorithms on physical platforms and discuss their applications. We discuss possible improvements in computational speed-up, resource saving and implementations on physical platforms. A qudit with a larger state space than a qubit can utilize the full potential of physical systems such as photon in Section 5.1, ion trap in Section 5.2, nuclear magnetic resonance in Section 5.3 and molecular magnet in Section 5.4. Finally, we give a summary of the qudit systems advantages and provide our perspective for the future developments and applications of the qudit in Section 6.

## Quantum Gates for Qudits

A *qudit* is a quantum version of *d*-ary digits whose state can be described by a vector in the *d* dimensional Hilbert space

where

This section gives a review of various qudit gates and their applications. Section 2.1 provides criteria for the qudit universality and introduces several fundamental qudit gate sets. Section 2.2 presents examples of qudit gates and illustrates their advantages compared to qubit gates. In the last section, Section 2.3, a quantitative discussion of the circuit efficiency is included to give a boundary of the number of elementary gates needed for decomposing an arbitrary unitary matrix.

### Criteria for Universal Qudit Gates

This subsection describes the universal gates for qudit-based quantum computing and information processing. We elaborate on the criteria for universality in Section 2.1.1 and give examples in Section 2.1.2.

#### Universality

In quantum simulation and computation, a set of matrices *U* of the Hilbert space *universality* not only applies to the qubit systems [47]; but can also be extended to the qudit logic [24, 39, 65, 102, 114, 164]. Several discussions of standards and proposals for a universal qudit gate set exist. Vlasov shows that the combination of two noncommuting single qudit gates and a two-qudit gate are enough to simulate any unitary *N* qudits [118]. They use the spectral decomposition of unitary transformations and involve a gate library with a group of continuous parameter gates. Brennen et al. [21] identify criteria for exact quantum computation in qudit that relies on the QR decomposition of unitary transformations. They generate a library of gates with a fixed set of single qudit operations and “one controlled phase” gate with single parameter as the components of the universal set. Implementing the concept of a coupling graph, they proved that by connecting the nodes (equivalently logical basis states) they can show the possibility of universal computation.

#### Examples of Universal Gate Sets

An explicit and physically realizable universal set comprising one-qudit general rotation gates and two-qudit controlled extensions of rotation gates is explained in this section [108]. We first define

as a transformation in the *d*-dimension that maps any given qudit state to

with

The *d*-dimensional phase gate is

which changes *θ* and ignores the other states, and

Each primitive gate (such as

which is a *d* substates

Now we work on an *n*-qudit state. The sufficiency of the gates *U*. By the representation theory, the unitary matrix *U* with *N* eigenvalues

with eigenoperators

Then the eigenoperators can be synthesized with two basic transformations as [118].

Here *N*-dimensional analogues of *j*th eigenstate to produce *j*th eigenphase *U*.

The second step is decomposing

which acts on the *m*-qudit space. It is proved in the appendix of Ref. 108 that each *x*). The last box contains

is universal for the quantum computation using qudit systems.

**FIGURE 1**. The schematic circuit of *m* controlling qudits. The two-qudit controlled gates is shown as the verticle lines.

One advantage of the qudit model (compared to the qubit model) is a reduction of the number of qudits required to span the state space. To explain this, we need at least *N*-dimensional system in qubits while in qudits we need *N* eigenoperators is needed and each can be decomposed with three rotations shown in Eq. 9. Deriving from the appendix of Ref. 108; *m* number of *L* in this decomposition method is

It is clear that there is an extra factor of *n* reduction in the gate requirement as the number scale of this method is

For qudit quantum computing, depending on the implementation platform, other universal quantum gate sets can be considered. For example, in a recent proposal for topological quantum computing with metaplectic anyons, Cui and Wang prove a universal gates set for qutrit and *qupit* systems, for a qupit being a qudit with *p* dimensions and *p* is an prime number larger than 3 [38]. The proposed universal set is a qudit analogy of the qubit universal set and it consists several generalized qudit gates from the universal qubit set.

The generalized Hadamard gate for qudits

where

The

The Pauli

with ω defined by Eq. 14 and the related

In general *p* is an odd prime.

The proposed gate set for the qutrit system is the sum gate

### Examples of Qudit Gates

In this section we introduce the qudit versions of many important quantum gates and discuss some of the gates’ advantages compared to their qubit counterparts. The gates discussed are the qudit versions of the

#### Qudit Versions $\mathbf{\pi}/8$ Gate

The qubit *T* has an important role in quantum computing and information processing. This gate has a wide range of applications because it is closely related to the Clifford group but does not belong to the group. From the Gottesman-Knill theorem [64] it is shown that the Clifford gates and Pauli measurements only do not guarantee universal quantum computation (UQC). The *d* dimensional qudit system, where, throughout the process, *d* is assumed to be a prime number greater than 2 [71].

To define the Clifford group for a *d*-dimensional qudit space, we first define the Pauli *Z* gate and Pauli *X* gate. The Pauli *Z* gate and Pauli *X* gate are generalized to *d* dimension in the matrix forms [11, 67, 124, 130].

for *ω* the *Z* gate is adding different phase factors to each basis states and that of the *X* gate is shifting the basis state to the next following state. Using basis states the two gates are

In general, we define the displacement operators as products of the Pauli operators,

where *x* and *y* in the exponent of τ, *X* and *Z*. This leads to the definition of the Weyl-Heisenberg group (or the generalized Pauli group) for a single qudit as [11, 67, 124, 130].

where *Clifford group* [124, 157];

A recursively defined set of gates, the so-called Clifford hierarchy, was introduced by Gottesman and Chuang as

for

The following derivations follow those in Ref. 71. The explicit formula for building a Clifford unitary gate with

is

The special case

can be shown. In the

and

With all the mathematical definitions at hand, we are ready to give an explicit form of the qudit

A straightforward application of Eqs 20 and 30 yields

As

We define

From Eqs 26 and 31 we see that the right-hand side of Eq. 32 is the most general form, and we note that

After canceling common factors of

or, equivalently, using Eq. 20,

From here, we derive the recursive relation

We solve for the

where all factors are evaluated modulo *d*. For example, with

so that

The diagonal elements of *d* and, consequently,

For the

The ninth root of unity Eq. 14 is

The qutrit version of

Then the general solution is

For example, choosing

The

This gate also plays an important role in the magic-state distillation (MSD) protocols for general qudit systems, which was first established for qutrits [6] and then extended to all prime-dimensional qudits [25].

#### Qudit SWAP Gate

A SWAP gate is used to exchange the states of two qudit such that:

Various methods to achieve the SWAP gate use different variants of qudit controlled gates [4, 58, 112, 131, 155, 158, 159] as shown in Figure 2A,B. The most used component of the SWAP gate is a controlled-shift gate

with a modulo *d* addition. Its inverse operation is

In some approaches, the operation

which outputs the modulo *d* complement of the input. These circuits are more complex and less intuitive then the qubit SWAP gate [58] because they are not Hermitian, i.e.,

**FIGURE 2**. **(A)** is the qudit SWAP circuit using **(B)** is the qudit SWAP circuits with the

One way to create a Hermitian version of the qudit CNOT uses the GXOR gate

However, this SWAP gate needs to be corrected with an

where

In the rest of this section, we present a Hermitian generalization of the qudit CNOT gate with a symmetry configuration and a qudit SWAP circuit with a single type of qudit gate as shown in Figure 4A [61]. Compared with all the previously proposed SWAP gate for qudit, this method is easier to implement since there is only one type of gate *d*-level qudits

where *d*. Notice that, for

**FIGURE 4**. **(A)** is the qudit SWAP gate with the **(B)** is the decomposing

The

The inverse QFT undoes the Fourier transform process and the inverse of

The full evolution of the

It is easy to show that

#### Simplified Qubit Toffoli Gate With a Qudit

The Toffoli gate is well known for its application to universal reversible classical computation. In the field of quantum computing, the Toffoli gate plays a central role in quantum error correction [35]; fault tolerance [43] and offers a simple universal quantum gate set combined with one qubit Hadamard gates [141]. The simplest known qubit Toffoli gate, shown in Figure 5, requires at least five two-qubit gates [125]. However, if the target qubit has a third level, i.e., a qutrit, the whole circuit can be achieved with three two-qubit gates [133].

**FIGURE 5**. Decomposing qubit Toffoli gate with the universal qubit gates. *H* is the Hadamard gate, *T* is the *S* is the phase gate.

A new qutrit gate *Z* gate, which is achieved with a CNOT gate between two Hadamard gates. The Hadamard gate here operating on the qutrit is generalized from the normal Hadamard gate operating on a qubit—it only works with the

**FIGURE 6**. The Simplified Toffoli gate. The first two lines represent two control qubits and the third line represents a target qutrit that has three accessible levels. The initial and final quantum states of the quantum information carrier are encoded in the *H* is the generalized Hadamard gate such that

This method can be generalized to *n*-qubit-controlled Toffoli gates by utilizing a single (

The previous method turns the target qubit into a qudit; another method simplifies the Toffoli gate by using only qudits and treating the first two levels of the qudit as qubit levels and other levels as auxiliary levels. The reduction in the complexity of Toffoli gate is accomplished by utilizing the topological relations between the dimensionality of the qudits, where higher qudit levels serve as the ancillas [89].

Suppose we have a system of *n* qudits denoted as

At the same time, the two-qubit *E* of ordered pairs

with

The set *E* describes an *n*-vertex-connected graph. Let *n*-vertex connected *tree* (acyclic graph). The main result is: the *n*-qubit Toffoli gate can be achieved with less number of operations if

where *n*-qubit Toffoli gate can be realized by *n*-qubit Toffoli gate by the properties and special operations of the tree in topology can be found in Ref. 89. The advantage of this scheme is the scalability and the ability to implement it for the multi-qubit controlled unitary gate

These

#### Qudit Multi-Level Controlled Gate

For a qubit controlled gate, the control qubit has only two states so it is a “do-or-don’t” gate. Qudits, on the other hand, have multiple accessible states and thus a qudit-controlled gate can perform a more complicated operation [46]. The *Muthukrishan-Stroud gate* (MS gate) for a qudit applies the specified operation on the target qudit only if the control qudit is in a selected one of the *d* states, and leaves the target unchanged if the control qudit is in any other *d* states on the control qudit [118].

To fully utilize the *d* states on the control qudit, people have developed the quantum multiplexer to perform the controlled *U* operations in a qudit system as shown in Figure 7, where the MS gate and shifting gates are combined to apply different operations to the target depending on different states on the control states [87]. Here we discuss the *multi-value-controlled gate* (

**FIGURE 7**. *d*-valued Quantum Multiplexer for the second qutrit and its realization in terms of Muthukrishan-Stroud gates (the control *U* operation that only act on one specific control state). The gate labeled *d*). Depending on the value of the top control qudit, one of

For a *d*-dimensional qudit system, a two-qudit multi-value-controlled gate is represented by a

where each

### Geometrically Quantifying Qudit-Gate Efficiency

In a quantum computer, each qudit can remain coherent for a limited amount of time (decoherence time). After this time, the quantum information is lost due to the outside perturbations and noises. In the computation process, quantum gates take certain amount of time to alter the states of the qudits. The decoherence time of a qudit state limits the number of quantum gates in the circuit. Therefore, we need to design more efficient algorithms and circuits. A method exists to do a general systematic evaluation of the circuit efficiency with the mathematical techniques of Riemannian geometry [126]. By reforming the quantum circuits designing problems as a geometric problem, we are able to develop new quantum algorithms or to exploring and evaluating the full potential of the quantum computers. This evaluation is able to generalized to qutrit systems, where the least amount of the gates required to synthesize any unitary operation is given [100].

To begin with, we assume that the operations done by the quantum circuit can be described by a unitary evolution *U* derived from the time-dependent Schrödinger equation *U* can be characterized by a cost function *U*.

Now we transform the problem of calculating a lower bound to the gate number to finding the minimal geodesic distance between the identity operation *I* and *U*. Instead of Pauli matrices for the qubit representation of the Hamiltonian, the qutrit version of Hamiltonian is expanded in terms of the Gell-Mann matrices. Here we give an explicit form of the Gell-Mann matrices representation in *d*-dimension [109] which is used for qutrit (where

Here, diag represents the diagonal matrix,

acts on the *s*-th qudit with

*t* qudits at sites

Now the Hamiltonian in terms of the Gell-Mann matrices (with the notation *σ*) can be written as

All coefficients *σ* goes over all possible one- and two-body interactions whereas, in *σ* goes over everything else. The cost function is

where *p* is a penalty cost by applying many-body terms. Now that the control cost is well defined, it is natural to form the distance in the space *n*-qutrit unitary operators with unit determinant. We can treat the function *g* as:

The distance *I* and *U* which is the minimum curve connecting *I* and *U* equals to the minimal length solution to the geodesic equation

where *K* is an arbitrary operator in

All lemmas backing up the final theorem have been proven in detail [100]; but the reasoning behind can be summarized in four parts. First let *p* be the three- and more-body items penalty. With large enough *p*, the distance *p*. Secondly, we have

where *n*-qutrit unitary operator *U* generated by

where *H* as an *n*-qutrit one- and two-body Hamiltonian, a unitary operator

and can be generated with at most

All these lemmas combined gives the final theorem for the qutrit system: for a unitary operator *U* in *c*. It is worth mentioning that for any groups of unitaries *U*, which is labeled by the number of qudits *n*, the final theorem shows a quantum circuit exists with a polynomial of *U* to arbitrary accuracy. Alternatively,a polynomial-sized quantum circuit exists if and only if the distance *n*.

With appropriate modification, the Riemannian geometry method can be used to ascertain the circuit-complexity bound for a qudit system [109]. In this scheme, the unitary matrix *ε*, each unitary *d* is the dimension of the qudit, *n* is the number of qudits and *N* is the number of the intervals that *U* by finding the shortest geodesic curve linking *I* and *U*. This provides a good reference for the design of the quantum circuit using qudits.

## Quantum Algorithms Using Qudits

A qudit, with its multi-dimensional nature, is able to store and process a larger amount of information than a qubit. Some of the algorithms described in this section can be treated as direct generalizations of their qubit counterparts and some utilize the multi-dimensional nature of the qudit at the key subroutine of the process. This section introduces examples of the well-known quantum algorithms based on qudits and divides them into two groups: algorithms for the oracle-decision problems in Section 3.1 and algorithms for the hidden Abelian subgroup problems in Section 3.2. Finally, Section 3.3 discusses how the qudit gates can improve the efficiency of the quantum search algorithm and reduce the difficulty in its physical set-up.

### Qudit Oracle-Decision Algorithm

In this subsection we explore the qudit generalizations of the efficient algorithms for solving the oracle decision problems, which are quite important historically and used to demonstrate the classical-quantum complexity separation [44, 45]. The oracle decision problems is to locate the contents we want from one of the two mutually disjoint sets that is given. We start in Section 3.1.1 with a discussion about a single-qudit algorithm that determines the parity of a permutation. In Section 3.1.2, the Deutsch-Jozsa algorithm in qudit system is discussed and its unique extension, the Bernstein-Vazirani algorithm is provided in Section 3.1.3.

#### Parity Determining Algorithm

In this section we review a single qutrit algorithm which provides a two to one speedup than the classical counterpart. This algorithm can also be generalized to work on an arbitrary *d*-dimensional qudit which solves the same problem of a larger computational space [62]. In quantum computing, superposition, entanglement and discord are three important parts for the power of quantum algorithms and yet the full picture behind this power is not completely clear [151].

Recent research shows that we can have a speedup in a fault tolerant quantum computation mode using the quantum contextuality [72]. The contextual nature can be explained as “a particular outcome of a measurement cannot reveal the pre-existing definite value of some underlying hidden variable” [92, 93]. In other words, the results of measurements can depend on how we made the measurement, or what combination of measurements we chose to do. For the qudit algorithm discussed below, a contextual system without any quantum entanglement is shown to solve a problem faster than the classical methods [62]. Because this qudit algorithm uses a single qudit throughout the process without utilizing any correlation of quantum or classical nature, it acts as a perfect example to study the sources of the quantum speed-up other than the quantum correlation.

The algorithm solves a black-box problems that maps *d* inputs to *d* outputs after a permutation. Consider the case of three objects where six possible permutations can be divided into two groups: *even* permutation that is a cyclic change of the elements and *odd* permutation that is an interchange between two elements. If we define a function

and the remaining three odd function are

The circuit for the single qutrit algorithm in a space spanned by

using *ω* as the cube root of unity Eq. 14. The process starts with state

and, similarly,

Hence, application of

**FIGURE 8**. Schematic view of the quantum circuit for the parity determining algorithm.

Generalizing to a *d*–dimensional qudit system,

In this scenario, a positive cyclic permutation maps

#### Qudit Deutsch-Jozsa Algorithm

Deutsch algorithm (with its origin in Ref. 44 and improved in Ref. 33 is one of the simplest examples to show the speed advantage of quantum computation. Deutsch-Jozsa algorithm is *n*-qubits generalization of the Deutsch algorithm. Deutsch-Jozsa algorithm can determine if a function *constant*, with constant output, or *balanced*, that gives equal instances of both outputs [125]. The process itself consists of only one evaluation of the function *N* qubits in the query register and one in the answer register where Bob applies the function to the query register qubits and stores the results in the answer register. Alice can measure the qubits in the query register to determine whether Bob’s function is constant or balanced. This algorithm makes use of the superposition property of the qubit and reduces the minimum number of the function call from

The Deutsch-Jozsa algorithm can be performed in the qudit system with a similar setup. Furthermore, with the qudit system, Deutsch-Jozsa algorithm can also find the closed expression of an affine function accurate to a constant term [53]. The *constant* and *balanced* function in the *n* dimensional qudit case have the following definition: “An *r*-qudit multi-valued function of the form

is *constant* when *balanced* when an equal number of the *n* elements in the co-domain” [53].

It can be shown that all of the affine functions of *r* qudits

can be categorized to either constant or balanced functions [53]. If all the coefficients *n* to a unique element *n*-nary Deutsch-Jozsa algorithm, another trivial lemma is needed: Primitive

The circuit of the Deutsch-Jozsa algorithm in qudits is shown in Figure 9. This algorithm of *r* qudits can both distinguish whether a function *x*-register and thus is lost during the measurement. The other coefficients of the affine function *x*-register at the output,

**FIGURE 9**. The Deutsch-Jozsa circuit in qudit system. The

A detailed derivation of the circuit has been shown [53]; but the reasoning is an analogy to the qubit version of the Deutsch-Jozsa algorithm. If the function *x*-register have null amplitudes. Therefore, if every *x*-register qudit yields

The Deutsch-Jozsa algorithm in the qudit system shares the same idea while enabling more applications such as determining the closed form of an affine function. Although this algorithm is mainly of theoretical interest, the *n*-nary version of it may have applications in image processing. It has the potential to distinguish between maps of texture in a Marquand chart since the images of which are encoded by affine functions [121]. This algorithm can also be modified to set up a secure quantum key-distribution protocol [121]. Other proposed Deutsch-Jozsa algorithms exist such as a method that makes use of the artificially allocated (subsystems) as qudits [88] and a generalized algorithm on the virtual spin representation [86].

#### Qudit Generalization of the Bernstein-Vazirani Algorithm

In Section 3.1.2 we have discussed an application of a qudit Deutsch-Jozsa algorithm (DJA): verify a closed expression of an affine function. This application is closely related to the Bernstein-Vazirani algorithm discussed in this section. Given an input string and a function that calculates the bit-wize inner-product of the input string with an unknown string, the Bernstein-Vazirani algorithm determines the unknown string [12]. This algorithm can be treated as an extension of the Deutsch-Jozsa algorithm.

The qudit generalization of the Bernstein–Vazirani algorithm can determine a number string of integers modulo *d* encoded in the oracle function [95, 119]. First we introduce a positive integer *d* and consider the problem in modulo *d* throughout. Given an *N*-component natural number string

we define

for

The oracle in the algorithm applies *x* and computes the result, namely, the number string

The input state *x* is chosen to be *N* control-qudits into their

and

for ω a root of unity Eq. 14. The component-wize Fourier transform of a string encoded in the state

where

We denote the Fourier transform of the

Now we introduce the oracle as the

where

By applying the

Finally, obtain the *N* qudits of *N* quantum state of

using a single query of the oracle function.

The Bernstein-Vazirani algorithm clearly demonstrates the power of quantum computing. It outperforms the best classical algorithm in terms of speed by a factor of *N* [95]. The qudit generalizations of the Bernstein-Vazirani algorithm helps us comprehend the potential of the qudit systems.

### Qudit Algorithms for the Hidden Abelian Subgroup Problems

Many of the widely used quantum algorithms such as the discrete Fourier transform, the phase estimation and the factoring fit into the framework of the hidden subgroup problem (HSP). In this section, we review the qudit generalization of these algorithms. The qudit Fourier transform is discussed in Section 3.2.1 and its application, the PEA is reviewed in Section 3.2.2. A direct application of these algorithms, Shor’s factoring algorithm performed with qutrits and in metaplectic quantum architectures is also introduced Section 3.2.2.

#### Quantum Fourier Transform With Qudits

The quantum Fourier transform algorithm (QFT) is realizable on a qubit system [125]. QFT, as the heart of many quantum algorithms, can also be performed in a qudit system [145, 165]. In an *N*-dimensional system represented with *n d*-dimensional qudits, the QFT,

into a new basis set [26]

For convenience, we write an integer *j* in a base-*d* form. If

and, if

The QFT acting on a state

This process can be realized with the quantum circuit shown in Figure 10, and the fully expanded expression of the product form is shown on the right side of the figure. The generalized Hadamard gate

The matrix representation of

In the circuit the

The black dots in the circuit are multi-value-controlled gates that apply *j* times for a control qudit in state

**FIGURE 10**. Quantum Fourier transform in qudit system. *d*-dimensional Hadamard gate and the expression of the **Eq. 99**. Resultant states are shown to the right.

The QFT developed in qudit system offers a crucial subroutine for many quantum algorithm using qudits. Qudit QFT offers superior approximations where the magnitude of the error decreases exponentially with *d* and the smaller error bounds are smaller [165]; which outperforms the binary case [34].

#### Phase-Estimation Algorithm With Qudits

With the qudit quantum Fourier transform, we are able to generalize the PEA to qudit circuits [26]. Similar to the PEA using qubit, the PEA in the qudit system is composed by two registers of qudits. The first register contains *t* qudits and *t* depends on the accuracy we want for the estimation. We assume that we can perform a unitary operation *U* to an arbitrary number of times using qudit gates and generate its eigenvector *r*.

The following derivations follow those in Ref. 26. For convenience, we rewrite the rational number *r* as

As shown in Figure 11A, each qudit in the first register passes through the generalized Hadamard gate

Then the

Note that the function of the controlled operation

Therefore, through a process called the “phase kick-back”, the state of the first register receives the phase factor and becomes

The eigenvalue *r* which is represented by the state

The whole process of PEA is shown in Figure 11B. To obtain the phase

**FIGURE 11**. **A)** The circuit for the first stage of the PEA. The qudits in the second register whose states represent *U* operations and the generated phase factors are kicking back to the qudits in the first register, giving the results to the right. **(B)** The schematic circuit for the whole stage of PEA. After the first stage of the PEA, inverse Fourier transform (

The PEA in qudit system provides a significant improvement in the number of the required qudits and the error rate decreases exponentially as the qudit dimension increases [129]. A long list of PEA applications includes Shor’s factorization algorithm [142]; simulation of quantum systems [1]; solving linear equations [69, 128]; and quantum counting [147]. To give some examples, a quantum simulator utilizing the PEA algorithm has been used to calculate the molecular ground-state energies [8] and to obtain the energy spectra of molecular systems [13, 41, 42, 84, 154]. Recently, a method to solve the linear system using a qutrit version of the PEA has been proposed [138]. The qudit version of the PEA opens the possibility to realize all those applications that have the potential to out-perform their qubit counterparts.

Shor’s quantum algorithm for prime factorization gives an important example of super-polynomial speed-up offered by a quantum algorithm over the currently-available classical algorithms for the same purpose [143]. The order-finding algorithm at the core of the factoring algorithm is a direct application of the PEA. With the previous discussion on the qudit versions of the quantum Fourier transform and phase estimation, we have the foundation to generalize Shor’s factoring algorithm to the higher dimensional qudit system. Several proposals for performing Shor’s algorithm on the qudit system, such as the adiabatic quantum algorithm of two qudits for factorization [166]; exist. This method makes use of a time-dependent effective Hamiltonian in the form of a sequence of rotation operators that are selected accoding to the qudit’s transitions between its neighboring levels.

Another proposal carries out a computational resource analysis on two quantum ternary platforms [17]. One is the “generic” platform that uses magic state distillation for universality [25]. The other, known as a metaplectic topological quantum computer (MTQC), is a non-Abelian anyonic platform, where anyonic braiding and interferomic measurement is used to achieved the universality with a relatively low cost [37, 38]. The article discusses two different logical solutions for Shor’s period-finding function on each of the two platforms: one that encodes the integers with the binary subspace of the ternary state space and optimizes the known binary arithmetic circuits; the other encodes the integer directly in the ternary space using the arithmetic circuits stemming in Ref. 16. Significant advantages for the MTQC platform are found compared to the others. In particular the MTQC platform can factorize an *n*-bit number with

### Quantum Search Algorithm With Qudits

The quantum search algorithm, also known as Grover’s algorithm, is one of the most important quantum algorithms that illustrates the advantage of quantum computing. Grover’s algorithm is able to outperform the classical search algorithm for a large database. The size of the computational space in an *n*-qubit system is a Hilbert space of

Since there is a practical limit for the number of working qubits, the working Hilbert space can be expanded by increasing the dimension of each carrier of information, i.e., using qudits and qudit gates. Several schemes of Grover’s quantum search with qudits have been proposed, such as one that uses the discrete Fourier transform as an alternative to the Hadamard gate [54] or another *d*-dimensional transformation [101] for the construction of the reflection-about-average operator (also known as the diffusion operator). In this section, an instruction on setting up Grover’s algorithm in the qudit system is reviewed as well as a proposal of a new way to build a quantum gate *F* that can generate an equal-weight superposition state from a single qudit state [79]. With the new gate *F*, it is easier to realize Grover’s algorithm in a physical system and improve the overall efficiency of the circuit.

Grover’s algorithm solves the unstructured search problem by applying Grover’s oracle iteratively as shown in Figure 12B. To construct the oracle, we build qudit gates to perform the oracle function *s* as compared to all the others. The logic behind the algorithm is to amplify the amplitude of the marked state *N* dimensional search space. In each step Grover’s oracle is executed one time. This oracle can be broken into two parts: (1) *Oracle query.* The oracle shifts the phase of the marked state

(2) *Reflection-about-average.* This operation is a reflection about a vector

It is constructed by applying the generalized Hadamard gate *H*, applying phase shift to *H* again. It is straightforward to show that

The two steps combined form Grover’s operator

**FIGURE 12**. **(A)** Circuit illustration for Grover iteration, *F* gate is the proposed qudit gate that transforms the single-qudit state **(B)** Schematic circuit illustration of the qudit quantum search algorithm.

Building Grover’s operator in a qudit system can be simplified both algorithmically and physically. The most important improvement can be achieved by replacing the Hadamard gate *H* with *F* which drives the single-qudit state

with *F* function can be realized by a single physical interaction in a multipod system easily. The multipod system consists of *d* degenerate quantum states

Then from the two-state solution, we can calculate the dynamics of the multipod [97].

**FIGURE 13**. Illustration of a qudit multipod linkage: the top is in the original basis and the bottom is in the Morris-Shore basis.

This method of building *F* minimizes the number and the duration of algorithmic steps and thus is fast to implement and, in addition, it also provides better protection against detrimental effects such as decoherence or imperfections. Due to its conceptual simplicity, this method has applications in numerous physical systems. Thus, it is one of the most natural and simplest realizations of Grover’s algorithm in qudits.

## Alternative Models of Quantum Computing With Qudits

The gate-based description of quantum computing is useful to establish principles of quantum computing with qudits, similar to the case for qubits. There are various approaches to quantum computing besides the gate-based model, such as the measurement-based [134]; adiabatic quantum computing [3, 55] and topological quantum computing [57]. Qudit versions of these approaches are barely explored to date, and we summarize the current status of these studies below.

### Measurement-Based Qudit Computing

Measurement-based quantum computing was introduced as an alternative approach to quantum computing whereby a highly entangled state, such as a cluster state [22] or its graph-state generalization [70]; is prepared and then computation is performed by sequential single-qubit measurements in bases that are determined by a constant number of previous measurement outcomes [123, 134]. Measurement-based quantum computing is appealing in settings where preparing a highly entangled many-qubit graph state is feasible, such as parallelized controlled-phase operations [134] or cooling to the ground-state of a special Hamiltonian [123].

Measurement-based qudit quantum computing is unexplored to date. Preparatory work on generalizing graph states, implicitly including the cluster-state special case, to qudit graph states has been reported [85]. Regarding implement, qudit-based approaches have only been reported for the error-correction aspect of measurement-based qubit quantum computing [82]. In this approach, the cluster state is envisioned as comprising qudits, with the high-dimensional nature of qudits serving to encode qubits for error correction. They propose continuous-variable realizations of a qudit cluster state in a continuous-variable setting [82].

### Adiabatic Qudit Computing

Adiabatic quantum computing approaches quantum computing by encoding the solution of a computational problem as the ground-state of a Hamiltonian whose description is readily obtained; the solution is obtained by preparing the ground state of a Hamiltonian whose ground-state is efficiently constructed and then evolving slowly, according to the adiabatic condition, into a close approximation of the ground state of the Hamiltonian specifying the problem [55]. The advantage of adiabatic quantum computing is evident in its natural correspondence to quantizing satisfiability problems [55]; and current efforts to exploit adiabatic quantum computing focus on quantum annealing, which is a quantum generalization of the simulated annealing metaheuristic used for non-quantum global optimization problems [40, 56, 83].

Quantum annealing is an important branch of quantum computing, particularly at the commercial level exemplified by D-Wave’s early and continuing work in this domain. As D-Wave researchers themselves point out, realistic solid-state devices treated as qubits are not actually two-level systems and higher-dimensional representations of the dynamics must be considered to model and simulate realistic solid-state quantum annealers. The effect of states outside the qubit space, namely the treatment of solid-state quantum annealing as qudit dynamics, has been studied carefully with conditions established for soundness of qubit approximations [5].

In fact the qudit nature of so-called superconduting qubits, i.e., the higher-dimensional aspects of the objects serving as qubits, is not just a negative feature manifesting as leakage error; remarkable two-qubit gate performance is achieved by exploiting adiabatic evolution involving avoided crossings with higher levels [10, 110] with this exploitation for fast, high-fidelity quantum gates extendable to three-qubit gates and beyond by exploiting intermediate qudit dynamics and avoided level crossings [160, 161]. Another suggestion for exploiting qudit dynamics concerns using a degenerate two-level system with the additional freedom perhaps improving the energy gap and thus increasing success probability [156].

A dearth of studies have taken place to date into qudit-based adiabatic quantum computing. The one proposal thus far concerns a quantum adiabatic algorithm for factorization on two qudits [166]. Specifically, they consider two qudits of possibly different dimensions, thus necessitating a hybrid two-qudit gate [39]. They propose a time-dependent effective Hamiltonian to realize this two-qudit gate and its realization as radio-frequency magnetic field pulses. For this model, they simulate factorization of each of the numbers 35, 21, and 15 for two quadrupole nuclei with spins 3/2 and 1, respectively, corresponding to qudit dimensions of 4 and 3, respectively.

### Topological Quantum Computing With Qudits

Topological quantum computing offers advantages over other forms of quantum computing by reducing quantum error correction overheads by exploiting topological protection. Some work has been done on topological quantum computing with qudits by proposing quantum computing with parafermions [49, 74].

Majorana fermions are expected to exhibit non-abelian statistics, which makes these exotic particles, or their quasiparticle analogue, sought after for anyonic quantum computing [90]. Majorana fermions can be generalized to *d* in the four-parafermion fusion space enables all single-qudit Clifford gates to be generated modulo phase terms [74].

Clifford gates do not provide a universal set of gates for quantum computing. A non-Clifford gate can be achieved for parafermions encoded into parafermion zero modes by exploiting the Aharonov-Casher effect, physically implemented by move a half-fluxon around the parafermionic zero modes. Combining this non-Clifford gate with the Clifford gates achieved by parafermion braiding yields a universal gate set of non-abelian quantum computing with qudits [49].

## Implementations of Qudits and Algorithms

The qubit circuit and qubit algorithm have been implemented on various physical systems such as defects in solids [27, 81, 120]; quantum dots [104, 127]; photons [113, 132]; super conducting systems [29, 31]; trapped ions [14, 15]; magnetic [7, 18, 32, 148] and non-magnetic molecules [30, 152]. For each physical representation of the qubit, only two levels of states are used to store and process quantum information. However, many quantum properties of these physical systems have more than two levels, such as the frequency of the photon [106]; energy levels of the trapped ions [91]; spin states of the nuclear magnetic resonance systems [48] and the spin state of the molecular magnetic magnets [115]. Therefore, these systems have the potential to represent qudit systems. In this section, we briefly review several physical platforms that have been used to implement qudit gates or qudit algorithms.

Although most of the systems have three or four levels available for computation, they are extensible to higher level systems and scalable to multi-qudit interactions. These pioneer implementations of qudit systems show the potential of future realization of the more powerful qudit quantum computers that have real-life applications.

### Time and Frequency Bin of a photon

Photonic system is a good candidate for quantum computing because photons rarely interact with other particles and thus have a comparatively long decoherence time. In addition, photon has many quantum properties such as the orbital angular momentum [9, 52]; frequency-bin [75, 76, 96, 107] and time-bin [73, 78] that can be used to represent a qudit. Each of these properties provides an extra degrees of freedom for the manipulation and computation. Each degree of freedom usually has dimensions greater than two and thus can be used as a unique qudit. The experimental realization of arbitrary multidimensional multiphotonic transformations has been proposed with the help of ancilla state, which is achievable via the introduction of a new quantum nondemolition measurement and the exploitation of a genuine high-dimensional interferometer [60]. Experimental entanglement of high-dimensional qudits, where multiple high-purity frequency modes of the photons are in a superposition coherently, is also developed and demonstrated [96].

Here we review a single photon system that has demonstrated a proof-of-principle qutrit PEA [106]. In a photonic system, there is no deterministic way to interact two photons and thus it is hard to build a reliable controlled gate for the photonic qudits. The following photonic system bypasses this difficulty via using the two degrees of freedom on a single photon—i.e., the time-bin and frequency-bin to be the two qutrits. The frequency degree of freedom carries one qutrit as the control register and the time degree of freedom carries another qutrit as the target register. The experimental apparatus consists of the well-established techniques and fiber-optic components: continuous-wave (CW) laser source, phase modulator (PM), pulse shaper (PS), intensity modulator (IM) and chirped fiber Bragg grating (CFBG). The device is divided into three parts [106]: 1) A state preparation part that comprises a PM followed by a PS and a IM that encodes the initial state to qudits; 2) a controlled-gate part that is built with a PM sandwiched by two CFBGs to perform the control-*U* operation; and 3) an inverse Fourier transformation comprising a PM and then a PS to extract the phase information. Note that the controlled-gate part can perform a multi-value-controlled gate that applies different operations based on the three unique states of the control qutrit. In the PEA procedure, eigenphases can be retrieved with

Here we provide an example for the statistical inference of the phase based on numerical data generated by the photonic PEA experiment just described. The two unitary operations used in the experimental setup are

with *ω* being the cube root of unity Eq. 14, and

In the experiment, photonic qutrits are sent through the control and target registers and the state of the control register qutrits is measured and counted to obtain the phase information.

Given the eigenphase ϕ of an eigenstate of the target register, the probability for the qutrit output state to fell into

Now let

The estimated phases for *π*), but, by repeating the experiment, the eigenvalue can be estimated from the statistical distribution of the results.

**TABLE 1**. Normalized photon counts and comparison of the true phase *ϕ* and the experimentally estimated phase **Eq. 110**) and **Eq. 111**) [106].

### Ion Trap

Intrinsic spin, an exclusively quantum property, has an inherently finite discrete state space which is a perfect choice for representing qubit or qudit. When a charged particle has spin, it possess a magnetic momentum and is controllable by external electromagneic pulses. This concept leads to the idea of ion trap where a set of charged ions are confined by electromagnetic field. The hyperfine (nuclear spin) state of an atom, and lowest level vibrational modes (phonons) of the trapped atoms serves as good representations of the qudits. The individual state of an atom is manipulated with laser pulse and the ions interact with each other via a shared phonon state.

The set-up of an ion trap qutrit system reviewed here can perform arbitrary single qutrit gates and a control-not gate [91]. These two kinds of gates form a universal set and thus can be combined to perform various quantum algorithms such as those discussed in Section 3. The electronic levels of an ion are shown in Figure 14. The energy levels

the effective Hamiltonian describing the ion in this system is

Knowing the Hamiltonian we are able to derive the evolution operator in the restricted three-dimensional space spanned by

where

The notation *g* and

This evolution operator can perform all kinds of the required coherent operations that are acting on any two of the logical states. It operates on the system and works essentially as a single qutrit gate. All kinds of transitions can be realized by manipulating the κ and

**FIGURE 14**. Electronic level structure of the trapped ion. The carrier of the quantum information is the qutrit states

Single qutrit gate alone is not sufficient to form a universal computational set, as we need a conditional two-qutrit gate or a two-qutirt controlled-gate to achieve universality. To define the conditional two-qutrit gate we need an auxiliary level

Here *a* is the annihilation operator and *ϕ* is the laser phase, and δ is the detuning. The Lamb-Dicke parameter is

This Hamiltonian governs the coherent interaction between qutrits and collective CM motion. With appropriate selection of effective interaction time and laser polarizations, the CM motion coupled to electronic transitions is coherently manipulated [91].

To complete the universal quantum computation requirements, we need to develop a measurement scheme. In this scheme, von Neumann measurements distinguishing three directions

### Nuclear Magnetic Resonance

Nuclear magnetic resonance (NMR) is an essential tool in chemistry and involves manipulating and detecting molecules’ nuclear spin states using radio-frequency electromagnetic waves [19]. Some technologies of this field are sophisticated enough to control and observe thousands of nuclei in an experiment. The NMR has the potential to scale up quantum computer to thousands of qudits [144].

In this section we review the implementation of a single-qudit algorithm that can determine the parity of a permutation on an NMR system [48]. The algorithm itself is the parity determining algorithm explained in Section 3.1.1. The molecule in this NMR setup is embedded in a liquid crystalline environment and the strong magnetic field is used to adjust the anisotropic molecular orientation. This adding a finite quadrupolar coupling term to the Hamiltonian which is as follows

where

Final states of the system can be derived from a single projective measurement. Pseudopure spin states act as approximation of effect of the system on an ensemble NMR quantum computer since it is impossible to do the true projective measurements [98]. The fidelity measurement of the experiment is given as

is used, where

Another set-up of the same algorithm treats a single quart [62]. The algorithm implementation is achieved using a spinâ€“

### Molecular Magnets

Molecular quantum magnets, also called the single-molecule magnets (SMM), provides another physical representation of qudits [115]. They have phenomenal magnetic characteristics and can be manipulated via chemical means. This enables the alternation of the ligand field of the spin carriers and the interaction between the SMM with the other units. As pointed out in one of the proposals, the nuclear spin states of the molecules, which have a long life-time, are used to store the quantum information. This information is read out by the electronic states. In the mean time, the robustness of the molecule allows it to conserve its molecular, electronic and magnetic characteristics at high temperatures [116].

As one of the SMMs, the single molecule

where

This measurement uses the technique of electro-migration. Initialization and manipulation of the four spin states of

Statistical analysis of the nuclear spin coherence time makes use of the spinâ€“lattice relaxation times by fitting the data for an exponential form

## Summary and Future Outlook of Qudit System

### Summary of the Advantages of Qudit Systems Compared to Qubit Systems

Throughout the article we discuss and review many aspects of the qudit systems such as qudit gates, qudit algorithms, alternative computation models and implementations. Most gates and algorithms based on qudits have some advantages over those for qubits, such as shorter computational time, lower requirement of resources, higher availability, and the ability to solve more complex problems. The qudit system, with its high-dimensional nature, can provide more degrees of freedom and larger computational space. This section summarizes the advantages of the qudit system compared to the qubit system.

Qudit gates have the advantage of a larger working Hilbert space which reduces the number of qudits needed to represent an arbitrary unitary matrix. In our discussion of universality in Section 2.1.2, the qudit method proposed by Muthukrishnan and Stroud’s has a *n* reduction in the gate requirement, where *n* is the number of qudits. By introducing qudits to the construction of some well-known gates such as the Toffoli gate, the elementary gate required are reduced from *k* is an integer that depends on the accuracy of the approximation and can be smaller than 6 [100].

For many of the physical systems such as photons [113, 132]; super conducting systems [29, 31]; trapped ions [14, 15]; magnetic [7, 18, 32, 148] and non-magnetic molecules [30, 152] there are usually more than two available physical states available for the applications. The qudit system has a higher efficiency utilizing those extra states than the qubit system. Also using the photonic system, we can perform the multi-level controlled gate (Section 2.2) which can perform multiple control operations at and same time and largely reduce the number of controlled gates requirement [106].

Other than computation, the qudit also has advantages in quantum communication as it possesses a higher noise resilience than the qubit [36]. The qudit system has a higher quantum bit error rate (QBER), which is a measure of resistance to the environmental noise or eavesdropping attacks, compared to the qubit system. The higher noise tolerance of the qudits helps to increase the secret key rate as it can be shown that the secret key rate increases as the Hilbert space dimensions increase at the same noise level [140]. Notice that in practical situation, the qudit system performed on each particular physical apparatus has varied amount of advantages than the qubit and there might be cases in which the high-dimensional states have a higher transmission distance [36]. This higher noise resilience of qudits is more advantageous if the qudits are entangled. The entanglement becomes more robust by increasing the dimension of the qudits while fixing their numbers. In other words, as the noise sources act locally on every system, increasing the dimension *d* will reduce the number of systems and thus reduce the effect of noise resulting in the robustness increase [103]. The increasing noise level tolerance as the qudit dimension increases can be shown on an photonic OAM system as an example of its implementation [51].

In summary the qudit system possesses advantages in the circuit design, physical implementation and has the potential to outperform the qubit system in various applications.

### Future Outlook of Qudit System

This review article introduces the basics of the high-dimensional qudit systems and provides details about qudit gates, qudit algorithms and implementations on various physical systems. The article serves as a summary of recent developments of qudit quantum computing and an introduction for newcomers to the field of qudit quantum computing. Furthermore we show the advantages and the potential for qudit systems to outperform qubit counterparts. Of course these advantages can come with challenges such as possibly harder-to-implement universal gates, benchmarking [80, 94, 117]; characterization of qudit gate [68, 136] and error correction connected with the complexity of the Clifford hierarchy for qudits [157].

Compared to qubit systems, qudit systems currently have received less attention in both theoretical and experimental studies. However, qudit quantum computing is becoming increasingly important as many topics and problems in this field are ripe for exploration. Extending from qubits to qudits uses in some mathematical challenges, with these mathematical problems elegant and perhaps giving new insights into quantum computing in their own right. Connections between quantum resources such as entanglement, quantum algorithms and their improvements, scaling up qudit systems both to higher dimension and to more particles, benchmarking and error correction, and the bridging between qudits and continuous-variable quantum computing [67] are examples of the fantastic research directions in this field of high-dimensional quantum computing.

## Author Contributions

All authors discussed the relevant materials to be added and all participated in writing the article.

## Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Acknowledgments

We would like to acknowledge the financial support by the National Science Foundation under award number 1839191-ECCS. BCS appreciates financial support from NSERC and from the Alberta government.

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Keywords: quantum information, quantum computing, qudit gates, qudit algorithm, qudit implementation

Citation: Wang Y, Hu Z, Sanders BC and Kais S (2020) Qudits and High-Dimensional Quantum Computing. *Front. Phys.* 8:589504. doi: 10.3389/fphy.2020.589504

Received: 03 August 2020; Accepted: 22 September 2020;

Published: 10 November 2020.

Edited by:

Marcelo Silva Sarandy, Fluminense Federal University, BrazilReviewed by:

Eduardo Duzzioni, Federal University of Santa Catarina, BrazilJun Jing, Zhejiang University, China

Copyright © 2020 Wang, Hu, Sanders and Kais. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Sabre Kais, kais@purdue.edu