Systematic construction of topological-nontopological hybrid universal quantum gates
based on many-body Majorana fermion interactions

A quantum computer is a promising next generation computer[1, 2, 3]. In order to execute any quantum algorithms, universal quantum computation is necessary[4, 5, 6]. There are various approaches to realize universal computation including superconductors[7], photonic systems[8], quantum dots[9], trapped ions[10] and nuclear magnetic resonance[11, 12]. The Solovay-Kitaev theorem dictates that only the Hadamard gate, the $\pi/4$ phase-shift gate and the CNOT gate are enough for universal quantum computation. These one and two-qubit quantum gates can be embedded to larger qubits straightforwardly in these approaches.

Braiding of Majorana fermions is the most promising method for topological quantum computation[13, 14, 15, 16, 17]. There are various approaches to materialize Majorana fermions such as fractional quantum Hall effects[18, 19, 16, 20], topological superconductors[21, 22, 23, 24, 25, 26, 27] and Kitaev spin liquids[28, 29]. However, it can generate only a part of Clifford gate[30, 31]. The entire Clifford gates are generated for two qubits but not for more than three qubits[31]. Furthermore, only the Clifford gates are not enough to exceed classical computers, which is known as the Gottesman-Knill theorem[32, 33, 34].

There are several attempts to make universal quantum computation based on Majorana fermions[30, 20, 35, 36, 24, 37, 38, 39, 40, 41]. In the Majorana system, it is necessary to construct logical qubits from physical qubits by taking a parity definite basis, because braiding preserves the fermion parity. It makes logical qubits nonlocal. It is a nontrivial problem to embed a nonlocal $M$-qubit quantum gate in the $N$-qubit system with $N>M$. Hence, even if the Hadamard gate, the $\pi/4$ phase-shift gate and the CNOT gate are constructed, it is not enough for universal quantum computation in the $N$-qubit system unless this embedding problem is resolved.

In this paper, we systematically construct various quantum gates for universal quantum computation by introducing $2N$-body interactions of Majorana fermions preserving the fermion parity. We have required the fermion parity preservation because it is beneficial to use the standard braiding process as much as possible due to its topological protection. By combining topological quantum gates generated by braiding and additional quantum gates generated by many-body interactions of Majorana fermions, topological-nontopological hybrid universal quantum computation is possible. It would be more robust than conventional universal quantum computation because the quantum gates generated by braiding are topologically protected. We systematically construct arbitrary Cⁿ-phase shift gates, the Hadamard gate, CⁿNOT gates and CⁿSWAP gates in the $N$-qubit system by this generalization.

Supplementary Materials are prepared for detailed analysis in the case of small qubits to make clear a general analysis for the $N$-qubit system.

Physical qubits and logical qubits: Majorana fermions are described by operators $\gamma_{\alpha}$ satisfying the anticommutation relations

The braid operator is defined by[14]

It satisfies $\mathcal{B}_{\alpha\beta}^{4}=1$ and there is a corresponding anti-braiding operator $\mathcal{B}_{\alpha\beta}^{-1}=\mathcal{B}_{\alpha\beta}^{3}$.

The qubit basis is defined by[14]

with $n_{\alpha}=0$ or $1$, where ordinary fermion operators are constructed from two Majorana fermions as

$2N+4$ Majorana fermions constitute $N+1$ physical qubits.

The braiding operation preserves the fermion parity $P_{\alpha\beta}\equiv i\gamma_{\beta}\gamma_{\alpha}$, where it commutes with the braid operator $\mathcal{B}_{\alpha\beta}$,

It means that if we start with the even parity state $\left|00\cdots 0\right\rangle_{\text{physical}}$, the states after any braiding process should have even fermion parity. Therefore, in order to construct $N$ logical qubits $\left|n_{N}\cdots n_{2}n_{1}\right\rangle_{\text{logical}}$, $N+1$ physical qubits $\left|n_{N+1}^{\prime}\cdots n_{2}^{\prime}n_{1}^{\prime}\right\rangle_{\text{physical}}$ are necessary[42, 43, 44]. There are $N!$ correspondences between the logical and physical qubits in general. However, we adopt the following unique correspondence. When the logical qubit $\left|n_{N}\cdots n_{2}n_{1}\right\rangle_{\text{logical}}$ is given, we associate to it a physical qubit $\left|n_{N}\cdots n_{2}n_{1}n_{0}\right\rangle_{\text{physical}}$ by adding one qubit $n_{0}$ uniquely so that $\sum_{\alpha=0}^{N}n_{\alpha}=0$ mod $2$. Alternatively, when a physical qubit $\left|n_{N}\cdots n_{2}n_{1}n_{0}\right\rangle_{\text{logical}}$ is given, we associate to it a logical qubit $\left|n_{N}\cdots n_{2}n_{1}\right\rangle_{\text{physical}}$ just by eliminating the qubit $n_{0}$. An example reads as follows,

This correspondence is different from those in the previous works[13, 43, 44, 45, 46, 47]. Accordingly, the detailed braiding process for quantum gates are slightly different from the previous ones[43, 44, 45, 46, 47].

$2N$-body interactions: A generic operator involving two Majorana fermions $\gamma_{\alpha}$ and $\gamma_{\beta}$ is expressed as $U_{\alpha\beta}=a_{1}+a_{2}\gamma_{\alpha}+a_{3}\gamma_{\beta}+a_{4}\gamma_{\beta}\gamma_{\alpha}$, since higher-order terms $\gamma_{\alpha}^{p}\gamma_{\beta}^{q}$ are absent for $p\geq 2$ and $q\geq 2$ because of the relations $\gamma_{\alpha}^{2}=\gamma_{\beta}^{2}=1$. Then, by imposing the parity conservation condition $\left[U_{\alpha\beta},P_{\alpha\beta}\right]=0$ with the fermion-parity operator $P_{\alpha\beta}=\gamma_{\beta}\gamma\alpha$, it is restricted to $U_{\alpha\beta}=a_{1}+a_{4}\gamma_{\beta}\gamma_{\alpha}$. Furthermore, the unitary condition $U_{\alpha\beta}U_{\alpha\beta}=U_{\alpha\beta}^{\dagger}U_{\alpha\beta}=1$ leads to the representation of $U_{\alpha\beta}$ in the form of

The choice $\theta=\pi/4$ corresponds to the braiding operation. In general, $\theta$ can take an arbitrary value.

This operator transforms the Majorana operators as

We show that non-Clifford gates are constructed based on them.

The two-body operation is realized by the unitary dynamics,

with

The four-body operation

is introduced[13] as an essential ingredient of universal quantum computation. We generalize it to the four-body operation $\mathcal{B}_{\alpha\beta\gamma\delta}^{\left(4\right)}\left(\theta\right)$ defined by

which keeps the parity

We use an abbreviation,

The relation

holds because of the coefficient $i$ in Eq.(13).

It is realized by the unitary dynamics,

with

Similary, we may define the $2N$-body operation by

It satisfies the unitary condition,

We also define

It is realized by the $2N$-body interaction of Majorana fermions

$N$ physical qubits: We consider the $2N$ Majorana fermion system. The explicit actions on $2N$ physical qubits are given by

for odd numbers and

for even numbers, where we have defined the rotation along the $z$ axis by

and

$N-1$ logical qubits: $N-1$ logical qubits are constructed from $N$ physical qubits based on the correspondence (6).

Local $z$ rotation is possible for any qubits as in

Local $x$ rotation is possible for any qubits as in

where we have defined the rotation along the $x$ axis by

and $\mathcal{B}_{\alpha\sim\beta}\left(\theta\right)\equiv\mathcal{B}_{\alpha,\alpha+1,\cdots\beta-1,\beta}\left(\theta\right)$. Accordingly, the Hadamard gate is embedded in arbitrary $N$ qubits by using the decomposition formula,

Next, we show that it is possible to construct any $2^{N}$ diagonal operators based on many-body Majorana interactions, and hence, an arbitrary Cⁿ-phase-shift gate is constructed. By applying the Hadamard gate, it is possible to construct the CⁿNOT gate. There are ${}_{N}C_{M}$ patterns of $2M$-body unitary evolutions in $2N$ physical qubits. By taking a sum, we have $\sum_{M=1N}^{N}C_{M}=2^{N}$ independent physical qubits. They produce $2^{N-1}$ independent logical qubits because there are complementary operators $\mathcal{B}_{\alpha}\left(\theta\right)$ and $\mathcal{B}_{\overline{\alpha}}\left(\theta\right)$ which produce the same logical qubits. The complementary operators are denoted as

where $\overline{\alpha}$ indicates a set of indices which is the complementary set of $\alpha$ in the indices $1\sim 2N$. For example, we consider the case for four qubits $N=4$, where eight Majorana fermions exists. The following different braiding operators $\mathcal{B}_{56}$ and $\mathcal{B}_{123478}$ give an identical logical quantum gate

where $\alpha=56$ and $\overline{\alpha}=123478$ with $N=4$. See the full list of the complementary braiding operators for four physical qubits in Eq.(S143) in Supplementary Material.

There are $2^{N-1}$ independent components in $N-1$ logical qubits. On the other hand, there are $2^{N-1}$ independent many-body Majorana operators. Hence, it is possible to construct arbitrary diagonal operators by solving the linear equation. They include the Cⁿ-phase shift gates. Using the relation

with $q_{p}=0,1$, the Cⁿ-phase shift gate is constructed as

where $\mathcal{B}_{\text{odd}}$ contains odd number of $\sigma_{z}$ operators for logical qubits, while $\mathcal{B}_{\text{even}}$ contains even number of $\sigma_{z}$ operators for logical qubits. By setting $\phi=\pi$, we obtain the CⁿZ gate.

For example, the CZ gate in three qubits are embedded as

where $U_{\text{CZ}}^{p\rightarrow q}$ indicates that the controlloed qubit is $p$ and the target qubit is $q$. The CC$\phi$ phase-shift gate acting on three logical qubits in given by

Especially, the CCZ gate is constructed as follows,

The Toffoli gate is constructed by applying the Hadamard gate to the CCZ gate as in

See Fig.1(a1). The Fredkin gate is constructed by sequential applications of three Toffoli gates as in

where $U_{\text{CZ}}^{\left(p,q\right)\rightarrow r}$ indicates that the controlloed qubits are $p$ and $q$ while the target qubit is $r$. See Fig.1(b1).

The CⁿNOT gate is constructed from CⁿZ gate as

where the Hadamard gate is applied to $n$th qubit. See Fig.1(a2).

The CⁿSWAP gate is constructed from the CⁿZ gate as

where $U_{\text{C}^{n}\text{NOT}}^{\overline{p}\rightarrow p}$ indicates that the target qubit is $p$ and the others are controlled qubits, where $\overline{p}$ indicates the complementary qubits of the qubit $p$. See Fig.1(b2).

As a result, an arbitrary phase-shift gate, the CⁿNOT gate and the Hadamard gate are constructed in any logical qubits, and hence, the universal quantum computation is possible based on many-body interactions.

Discussions: We have analyzed the embedding problem inherent to the Majorana system, and shown that universal quantum computation is possible by introducing many-body interactions of Majorana fermions. Especially, the Cⁿ-phase shift gate, the CⁿNOT and the CⁿSWAP gate are systematically constructed, which are the basic ingredients of universal quantum computation based on the Solovay-Kitaev theorem. Although it was previously pointed out that four-body interactions of Majorana fermions are enough for universal quantum computation[13], we have shown that the four-body interactions are not enough but $2N$-body interactions are necessary.

The proposed quantum gates based on many-body interactions of Majorana fermions are not topologically protected because they are not generated by the standard braiding operations. By combining topological quantum computation based on braiding of Majorana fermions and nontopological quantum computation based on many-body interaction of Majorana fermions, topological-nontopological hybrid universal quantum computation is possible. It would be more robust than conventional universal quantum computation because many of quantum gates generated by braiding are topologically protected.

Recently, quantum simulation on Majorana fermions is studied in superconducting qubits[48, 49, 50]. It will be possible to realize many-body interactions of Majorana fermions in near future.

This work is supported by CREST, JST (Grants No. JPMJCR20T2) and Grants-in-Aid for Scientific Research from MEXT KAKENHI (Grant No. 23H00171).