Draft:Qudits and Qutrits

In classical computing, information is stored in bits, which can exist in one of two states: $0$ or $1$. Quantum computing, however, leverages the principles of quantum mechanics to allow for more complex information storage and processing. The basic unit of quantum information is the qubit, which exists in a superposition of both states.

Qudits are multi-level quantum systems that generalize the concept of a $2$-level qubit. While a qubit operates in a two-dimensional Hilbert space, a qudit operates in a $d$-dimensional Hilbert space, where $d$ is an integer greater than $2$. A qutrit is a specific case of qudit with $d$ equal to $3$, meaning it can exist in a superposition of three orthogonal states.

Qudits represent significant advancements in the field of quantum computing by extending the capabilities beyond traditional binary systems. They provide a larger state space for storing/encoding more information; processing more information; and doing multiple control operations in a simultaneous fashion. Their ability to encode more information within single units provides numerous advantages including reduced circuit complexity, simplified experimental setup, enhanced error resilience and enhanced algorithm efficiency. Some have even proposed advantages in error correction.

Like qubits, qudits can be implemented across various physical platforms, including:

• Trapped Ions: Ions confined in electromagnetic traps can be manipulated to represent qudits.

• Photonic Systems: The properties of photons, such as their polarization or orbital angular momentum, can be utilized to encode qudit states.

• Spin Systems: Particles with intrinsic spin greater than $1/2$ have multiple accessible states, making them suitable for qudit representation.

Qudits are also utilized in various classes of quantum algorithms such as non-abelian sungroup problems (e.g. phase-estimation algorithm), decision problems (e.g. Deutsch-Jozsa algorithm), search problems (e.g. Grover's algorithm).

Physical Realization of Qudits

As mentioned earlier, qudits can be physically realized using different systems including but not limited to neutral atoms (e.g. Rydberg atom), single molecular units, transmon. In the following are more details on two of the approaches.

Trapped Ions

Energy level diagram of a $^{40}\mathrm {Ca} ^{+}$ ion, illustrating electronic states, Zeeman sublevels, and laser-driven transitions used for realizing qudits.

Trapped-ion system is one of the first platforms to be proposed for quantum computing and has been demonstrated to show a quantum volume of $32768$ in the 15-qubit H1-1 processor. Quantinuum claimed at the time of publishing (2023), that it was the highest recorded in the industry. Some of the features of trapped ions are long coherence times, high-fidelity gates and essential all-to-all connectivity. However, there is the challenge of the scalability to large enough numbers of qubits without the decrease of the gate fidelities. But interestingly, the multilevel can be leveraged to achieve $d$-level quantum systems. One common approach is to encode multiple qubits in a single qudit; another approach is to substitute ancilla qubits with additional qudit levels, in multiqubit gate decompositions.

Typically in trapped-ion systems, quantum information is encoded in metastable states of ions; these ions are coupled by optical or microwave (mw) fields to perform quantum operations. The ions are directly laser cooled and loaded into traps; the transition between hyperfine components of the ground state are widely used as a microwave qubit. Such qubit can be readily initialized by optical pumping to a single Zeeman sublevel, and this can be extended to the case of qudits. For example, the Zeeman structure of the $729{\textit {nm}}$ optical transition in Calcium ($^{40}Ca^{+}$) ion was successfully used to encode a qudit up to $d=7$. The electronic ground state, $S_{\frac {1}{2}}$ and metastable state, $D_{\frac {5}{2}}$ are used to encode quantum information. A powerful magnetic field is used to split the ground state into two Zeeman sublevels ($m=\pm 1/2$), and the excited state into six Zeeman sublevels ($m=\pm 5/2,\pm 3/2,\pm 1/2$) such that each $^{40}\mathrm {Ca} ^{+}$ natively supports a qudit with eight levels (See the figure on the right). Similarly, all six Zeeman sublevels of both upper and lower levels of the Ytterbium ion ($^{171}Yb^{+}$) have been proposed for encoding $d=6$ qudits; although the researchers demonstrated the proof-of-principle by encoding $d=4$ qudits.

Photons

The photon is a natural candidate for encoding qudits. This is due to its various degrees of freedom (DOFs) i.e it is intrinsically multidimensional; the photon has been demonstrated to reliably encode qudits with its various DOFs such as orbital angular momentum (OAM) , time-bin , frequency-bin , spatial modes , and hybrid time-frequency bin encoding. For example, an experiment successfully encoded the $d=5$ qudit in the full-field spatial modes of photons. In this case, a single-photon source was used to generate photons that serve as carriers of quantum information. These photons were then prepared in Laguerre-Gauss (LG) modes to encode the qudit states. Another setup realized a qutrit through time-frequency encoding. They sent a continuous-wave (CW) laser source operating in the C-band through a phase modulator (PM1) driven at $18GHz$, thereby, creating a total number of $\approx 10$ frequency bins with a spacing of $18\;GHz$. After which they programmed a pulse shaper (PS1) to filter out everything except three equiamplitude frequency bins, now with a frequency spacing ($\Delta f$) of $54\;GHz$. The resulting coherent states are used as input rather than the true single photons. To prepare the target qutrit state, they used an intensity modulator (IM) driven by an arbitrary waveform generator (AWG), to carve out three narrow time bins each with a $6\,ns$ spacing, a $24\,ns$ repetition period, and a full width at half maximum of $\approx 0.2\,ns$ which broadens the frequency-bin line-width to $2.2\,GHz$.

Mathematical Framework

Qudits

A qudit is a quantum analog of $d$-ary digits represented as a vector in a $d$-dimensional Hilbert space, ${\mathcal {H}}_{d}$. This space is spanned by $d$ orthonormal basis vectors, $\mid 0\rangle$, $\mid 1\rangle \;\ldots$, $\mid d-1\rangle$. The general state of a qudit ($\psi \in {\mathcal {C}}^{d}$) is represented as:

$$\mid \psi \rangle =\sum _{k-0}^{d-1}\alpha _{k}\mid k\rangle ,$$

where $\alpha _{k}\in \mathbb {C}$ satisfy the normalization condition, $\sum _{k=0}^{d-1}|\alpha _{k}|^{2}=1$.

Qutrits

A qutrit is a specific instance of a qudit ($d=3$), capable of existing in a superposition of three orthogonal states: $\mid 0\rangle$, $\mid 1\rangle$, and $\mid 2\rangle$. The state of a qutrit can be mathematically represented as a vector in a three-dimensional complex Hilbert space, ${\mathcal {H}}_{3}$ and can be expressed as:

$$\mid \psi \rangle =\alpha _{0}|0\rangle +\alpha _{1}|1\rangle +\alpha _{3}|2\rangle ,$$

where $\alpha _{0},\alpha _{1},\alpha _{3}\in {\mathcal {C}}$ are complex coefficients that satisfy the normalization condition, $|\alpha _{0}|^{2}+|\alpha _{1}|^{2}+|\alpha _{3}|^{2}=1.$

Quantum Gates for Qudits and Qutrits

The qudit, just like its qubit counterpart, is transformed by unitary matrices called gates. And again like for qubits, the idea of universality of quantum gates applies to qudits. A set of matrices $U_{k}$ can be called the set of universal quantum gates if any combination of its elements can be used to approximate any arbitrary unitary transformation of the Hilbert space. Some examples of qutrit gates are:

• Hadamard Gate: The Hadamard gate for a qudit transforms the basis states as follows:

$$H_{d}|j\rangle ={\frac {1}{\sqrt {d}}}\sum _{k=0}^{d-1}w^{jk}|i\rangle {\text{ , }}j\in \{0,1,2,\ldots ,d-1\},\;w:=e^{2\pi i/d}.$$

The case of a qutrit (denoted $H_{3}$) is:

$$H_{3}|j\rangle ={\frac {1}{\sqrt {3}}}\sum _{k=0}^{2}e^{2\pi ijk//3}|k\rangle .$$

• SUM Gate: The $SUM_{d}$ generalizes the CNOT for a qudit

$$SUM_{d}|i,j\rangle ==|i,i+j({\text{mod }}d),$$

${\text{where }}i,j\in \{0,1,2,\ldots d-1\}$. So, for a qutrit, the CNOT gate is

$${\text{SUM}}_{3}|ij\rangle =\mid i,i+j\;{\textbf {mod}}\,3\,\rangle .$$

• Controlled-Phase (CZ) Gate: This is constructed by using entangling interaction. It is maximally entangled and a member of the two-qutrit Clifford group.

$$U_{CZ}=\sum _{ij\in \{0,1,2\}^{2}}w^{ij}|ij\rangle \langle ij|,$$

where $w^{ij}=e^{2i\pi /3}$ is the $3rd$ root of unity for a qutrit. It would be the $d$th root of unity in for generalized qudit.

For qudits, similar gates can be defined with appropriate modifications based on the dimensionality $d$.

Advantages of Qudits

Increased Information Density

One of the primary advantages of using qutrits and qudits over qubits is their ability to encode more information within a single unit i.e they have higher information coding capacity. This is because they provide a larger state space.

• A single qubit can represent two states ($0{\text{ or }}1$). For $n$ qubits, the state would reside in the tensor product space ${\mathcal {H}}_{2^{n}}={\mathcal {H}}_{2}^{\otimes n}$. This space has $2^{n}$ orthonormal basis states. You could say that $n$ qubits encode $2^{n}$ classical bits.
• But for a qudit, we have a Hilbert space ${\mathcal {H_{d^{n}}}}={\mathcal {H}}_{d}^{\otimes n}$ with dimension $d^{n}$. The space has $d^{n}$ orthonormal basis. The state space of $n$ qudits grows exponentially with $d$, offering a much larger computational state space than qubits for $d>2$.

This increased dimensionality allows for more efficient algorithms and potentially reduces the number of required operations in quantum computations.

Requirement for Spanning the State Space

The qudit model (compared to the qubit model) requires less number of qudits to span the state space. Consider the case of an $N$-dimensional system; we would need at least $n_{1}=log_{2}N$ qubits or $n_{2}=log_{d}N$ qudits to represent the system depending on which model. There is a binary-ternary compression factor of $k={\frac {n_{1}}{n_{2}}}=log_{2}d$ for the qudit system ($log_{2}3$ for qutrit). Equivalent constructions using Muthukrishnan and Stroud's method will require qubits and qudits gates on the scale of ${\mathcal {O}}(n_{1}^{2}N^{2})$ and ${\mathcal {O}}(n_{2}^{2}N^{2})$ respectively; showing that we can get a $(log_{2}d)^{2}$ scaling advantage in the qudit case over the qubit case. In summary, you would need fewer qudits for higher dimensions than you would qubits. A consequence of this is reduced overall hardware requirement.

Reduced Circuit Complexity

Qudits can simplify circuit designs by reducing the number of gates needed to perform certain operations. For example, algorithms that require multiple qubits may be executed using fewer qudits due to their larger state space. This reduction in complexity can lead to faster computations and lower error rates.

Enhanced Error Resilience

The use of higher-dimensional systems like qutrits may also enhance error resilience in quantum computations. By spreading information across multiple levels rather than confining it to just two (as with qubits), it becomes possible to mitigate certain types of errors that could affect computational outcomes.

Applications in Quantum Computing

Comparing Circuit Complexity for Using Qubits and Qudits in QFT. (a) The upper diagram shows the QFT requirement of only 4 qudits (d = 6) and 10 gates. (b) The lower diagram shows the QFT requirement of 10 qubits (d = 2) and 55 gates

Quantum Algorithms

Qudits (and by extension, qutrits) have been explored in various quantum algorithms that benefit from their extended state space. One of such, and a very important one as that is the quantum Fourier transform (QFT). It is heavily relied upon in quantum phase estimation, order finding, fast factorisation and Shor's algorithm. QFT in $n$-qudits quantum system can be denoted as some function, $F(d,N)$ that transforms some state $|j\rangle$ into $F(d,N)|j\rangle$,

$$F(d,N)|j\rangle ={\frac {1}{\sqrt {N}}}\sum _{k=0}^{N-1}e^{2\pi i\cdot j\cdot k/N}|k\rangle ,$$

where $N=d^{n}$ and $d,n\in {\mathcal {Z}}$ denote the dimension of each qudit and number of qudits, respectively. The set, $\{|0\rangle ,|1\rangle ,\ldots ,|N-1\rangle \}$ forms the computational basis. Given that $N=d^{n}$ and $j\in {\mathcal {Z}},0\leq j\leq d^{n}-1$, the Fourier transform of $Z_{d^{n}}$ can be defined as a linear operator that acts on $|j\rangle$ such that

$$F(d,d^{n})|j\rangle ={\frac {1}{\sqrt {d^{n}}}}\sum _{k=0}^{d^{n}-1}e^{2\pi i\cdot j\cdot k/d^{n}}|k\rangle .$$

Let us for the sake of convenience, write $j$ in a base-$d$ form such that if $j>1$, then

$$j=j_{1}j_{2}\ldots j_{n}=j_{1}d^{n-1}+j_{2}d^{n-2}+\ldots +j_{n}d^{0}.$$

But if $j<1$,

$$j=0\cdot j_{1}j_{2}\ldots j_{n}=j_{1}d^{-1}+j_{2}d^{-2}+\ldots +j_{n}d^{-n}.$$

Now, we can derive the Fourier transform in product form:

$${\begin{aligned}F(d,d^{n})|j\rangle &={\frac {1}{\sqrt {d^{n}}}}\sum _{k=0}^{d^{n}-1}e^{2\pi i\cdot j\cdot k/d^{n}}|k\rangle \\&={\frac {1}{d^{\frac {n}{2}}}}\sum _{k_{1}=0}^{d-1}\ldots \sum _{k_{n}=0}^{d-1}e^{2\pi ij(\sum _{l=1}^{n}k_{l}d^{-l})}|k_{1}k_{2}\ldots \rangle \\&={\frac {1}{d^{\frac {n}{2}}}}\sum _{k_{1}=0}^{d-1}\ldots \sum _{k_{n}=0}^{d-1}\bigotimes _{l=1}^{n}e^{2\pi ijk_{l}d^{-l}}|k_{l}\rangle \\&={\frac {1}{d^{\frac {n}{2}}}}\bigotimes _{l=1}^{n}\left[\sum _{k_{l}=0}^{d-1}e^{2\pi ijk_{l}d^{-l}}|k_{l}\rangle \right]\end{aligned}}.$$

This can easily be realized in a quantum circuit through a combination of the generalized Hadamad gate $H^{d}$ and a linear combination of controlled-phase gates, $R_{k}^{d}$ (See circuit diagram on the right). To understand the advantage that we can get from using qudits, consider that the first term in the transform requires one $H$ gate and $n-1\;R_{k}$ gates; the next term requires one $H$ gate and $n-2\;R_{k}$ gates, and so on. So that the number of gates required is:

$${\text{Number of gates}}=n+(n-1)+(n-2)+\ldots +1={\frac {n(n+1)}{2}}.$$

As an illustration, assume we need to represent a Hilbert space with $N=1024$ dimensions ($2^{10}=1024$).

• Using Qubits ($d=2$):
  • Number of qubits: $n=\log _{2}(1024)=10$ qubits.
  • Number of gates: ${\frac {10\times 11}{2}}=55{\text{ gates}}$.

• Using Qudits ($d=6$):
  • Number of qudits: $m=\log _{6}(1024)\approx 3.86$. Rounding up, $m=4$ qudits (since we cannot use fractional qudits).
  • Number of gates: ${\frac {4\times 5}{2}}=10{\text{ gates}}$.

• Savings from ( d = 6 )
  • Units: Using qudits reduces the number of quantum units from $n=10$ to $m=4$, saving $60\%$.
  • Gates: Using qudits reduces the number of gates from $55$ to $10$, saving $\approx 82\%$.

It is already obvious that we would have a less complex circuit with qudits than with qubits. Furthermore, because the larger dimension, quantum Fourier transform in qudit system is usually more efficient and accurate than qubit systems.

Some other applications of qudits include the strengthening of the Bell nonlocality; closing the detection loophole in Bell experiments ; high-capacity noise-resilient quantum cryptography and less resource overhead in quantum error correction.

External links

• Quantum Volume reaches 5 digits in Quantinuum
• Quantum 101 in Institute of Quantum Computing, Waterloo

See Also

• Quantum Gates
• Quantum Fourier Transform
• Alonso, Guillermo (9 May 2023). "Qutrits and quantum algorithms". PennyLane Demos.

Further Reading

• Wang, Yuchen; Hu, Zixuan; Sanders, Barry; Kais, Sabre (2020). "Qudits and High-Dimensional Quantum Computing" (PDF). Frontiers in Physics. 8:589504. doi:10.3389/fphy.2020.589504.

• Barnett, Stephen (2009). Quantum Information. Oxford University Press. ISBN 978-0198527633.

• Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum computation and quantum information (10th anniversary ed.). Cambridge: Cambridge university press. ISBN 978-1-107-00217-3.