Exact and approximate continuous-variable
gate decompositions

For a quantum computer to be able to execute an algorithm, often defined by a product of high-level unitary transformations [1, 2, 3, 4, 5, 6], the operations of the algorithm must be translated into the small set of logical gates directly built into the hardware. These built-in operations are the quantum computing analog to the logical gates which are hardwired into the transistors of a classical computer. In order to program a quantum computer to execute an algorithm, a combination of the built-in elementary gates must be found that can reproduce the operation of the algorithm. Certain sets of gates exist such that any arbitrary unitary operation can be expressed as a finite product of gates from the set, to any desired precision [7, 8, 9, 10, 11, 12]. Due to their ability to express any given transformation, these are referred to as universal gates sets. Thus, in order to program a quantum computer, a method is required to decompose any given unitary in terms of elements of the universal gate sets. Ideally, a decomposition method will allow for the implementation of an algorithm with high precision and as few gates as possible. Gate decomposition methods and techniques have been broadly studied, especially in the discrete variable (qubit) domain [7, 10, 13, 14, 15].

The qubit model of quantum computing uses two-level quantum systems that act as the quantum analog of bits on a classical computer. The logical gates acting on qubits can be represented by $2\times 2$ unitary matrices, and the process of translating general operations into the logical gates of this model, or gate decomposition, has been well studied. For example, the Solovay-Kitaev theorem [15, 5], stated informally tells us that any single-qubit operation can be approximated to high precision using a sequence of only a few logical gates. Other results have also been discovered which strengthen decomposition techniques in qubit systems by allowing for less logical gates to be needed, or the resulting decomposition to be of a greater precision. These techniques are used for single-qubit operations [16, 17, 18, 19] as well as general multi-qubit operations [6].

In this manuscript we focus on gate decomposition techniques for continuous-variable (CV) quantum computers, which have not been as extensively studied. In the CV model of quantum computing, registers are infinite-dimensional quantum systems called qumodes (the infinite-dimensional generalization of qubits) and physically correspond to quantum harmonic oscillators. The logic gates in this model are unitary operators which act on the infinite-dimensional Hilbert space [11, 20, 21, 22, 1, 2, 3, 4]. CV quantum computing using quadrature operators to form logic gates was first proposed by Lloyd and Braunstein [11]. The logic gates in that case were formed using Hamiltonians which were polynomial in the quadrature operators. These gates allow one to express analog computation using quantum-mechanical operations, and moreover, by constructing a universal set one can construct any gate generated by a Hamiltonian that is a polynomial of finite order in the quadrature operations, like the ones typically found in the simulation of bosonic systems [23, 2]. The problem of finding optimal ways to decompose an arbitrary gate into a product of gates from a given universal set is precisely the focus of this manuscript. The study of this problem also finds application when continuous degrees of freedom are used to encode error-correctable qubits, since the logical operations of the logical qubits need to be ultimately executed as continuous-variable gates [24, 25, 26, 27].

We have structured our manuscript as follows: in Sec. 2 we set up some basic notations. Then in Sec. 3 we examine a comprehensive method for decomposing arbitrary CV transformations using commutator identities introduced by Sefi and van Loock in Ref. [13]. This method is approximate, meaning that the final decomposition only implements the desired operation up to a certain error. The error is decreased by repeating the sequence of gates in the decomposition [28, 13, 29]. Exact decompositions which do not rely on Trotterization (see next section) are also known but are not universally applicable [1, 13, 3]. We first study exact decompositions of quadratic Hamiltonians in Sec. 4.1 using their symplectic representation [30, 31]. Then, in Sec. 4.2 we study exact decomposition of higher than quadratic polynomials of mutually commuting operators [14, 32]. These methods are further extended and generalized in Sec. 4.3. In Sec. 5 we examine another approximate method which relies on using squeezing and displacement operations to assemble a single mode cubic-phase gate starting from a Kerr gate [33]. Finally we provide a comparison of the methods as well as some discussion in Sec. 6.

Registers in the CV model are quantum harmonic oscillators with corresponding creation and annihilation operators $\hat{a}^{\dagger}_{j}$ and $\hat{a}_{j}$, where the subscript refers to the mode they act upon. For a system with $M$ modes, these operators satisfy the bosonic commutation relations $[\hat{a}_{j},\hat{a}^{\dagger}_{k}]=\delta_{jk}$, and $[\hat{a}_{j},\hat{a}_{k}]=[\hat{a}^{\dagger}_{j},\hat{a}^{\dagger}_{k}]=0$. An equivalent operator description of a bosonic system uses the quadrature operators $\hat{x}$ and $\hat{p}$, which can be represented in terms of the annihilation and creation operators as

with commutators $[\hat{x}_{j},\hat{x}_{k}]=[\hat{p}_{j},\hat{p}_{k}]=0$ and $[\hat{x}_{j},\hat{p}_{k}]=i\hbar\delta_{j,k}$. For convenience we group the operators as follows,

This notation allows us to write the commutation relations as $[\hat{r}_{j},\hat{r}_{k}]=i\hbar\Omega_{j,k}$ where

is the so-called symplectic form, and $\mathbb{I}_{M}$ is the $M$-dimensional identity matrix.

As mentioned previously, a universal gate set is a collection of gates such that any arbitrary unitary operation can be expressed as a finite sequence of gates from the set, to any desired precision. The CV decomposition techniques examined in later sections focus on the universal set specified by the gates

where $\theta$, $t_{1}$, $t_{2}$, $t_{3}$, and $\tau$ are real parameters. This particular universal set is often chosen for mathematical convenience. Gates generated by a Hamiltonian that is at most quadratic are referred to as Gaussian operations. In the universal set above the rotation operator (5a), pure-momentum displacement (5b), quadratic-phase (5c), and the multi-mode controlled-phase gate (5e) are all Gaussian. These operations are implemented experimentally with linear optics.

The cubic phase gate (5d), which is the only non-Gaussian operator in the universal set, provides the non-linearity needed to allow for decompositions of arbitrarily higher order. This gate can be realized either by using optical nonlinearities [33] or measurements [27, 34, 35, 36, 37, 38, 39]. For example, in [27] a photon counting measurement is used to introduce the higher order non-linearity. In this case the full implementation involves a displaced two-mode squeezed state for which $R^{\dagger}\hat{n}R$ (photon counting in a rotated basis) is measured on one arm. The desired cubic operation is then collapsed onto the second unmeasured mode. In [36] repeat-until-success photon subtractions and Gaussian operations are used, and in [39] quadrature detection for feed-forward manipulation of parameters produces the necessary nonlinear interaction. The cubic phase gate can also be realized in the microwave domain using the Josephson nonlinearity [40].

A rotation operation by $\theta=\frac{\pi}{2}$ is referred to as the Fourier transform gate [41, 42]. This operation maps between the quadrature operators $\hat{x}$ and $\hat{p}$ and is a cheap gate to implement experimentally.

An equivalent universal set is one where the non-Gaussian gate $V_{j}(t_{3})$ is replaced by the Kerr gate $\exp\left(it_{3}(\hat{a}^{\dagger}\hat{a})^{2}\right)$. In general, the non-Gaussian gates in the set can be replaced with any other and the set will retain universality. The reason why the cubic phase gate is chosen in the universal set here, as opposed to a Kerr gate, is due to mathematical convenience in the exact decompositions. As well, at the moment it seems as though gates with a mixture of $\hat{x}$ and $\hat{p}$ in the same mode cause problems for these same methods. This is because the polynomials used to create the desired powers in the decompositions will no longer be able to be constructed with unitary conjugation. Finally, note that in Sec. 5 we examine in detail a method derived in Ref. [33] where a driven-Kerr Hamiltonian is used to synthesize a cubic phase gate by unitary conjugation with squeezing and displacement gates.

One can also remove one of the Gaussian gates by showing that it can be decomposed using other elements in the set [13]. In both of these cases the choice is often made such that the gates in the set are not costly to implement experimentally. If the decomposition of one gate in the set includes multiple uses of another harder to implement gate, then it might be beneficial to still include it. For example, the quadratic phase gate can be decomposed in terms of other gates, but its decomposition includes multiple uses of the cubic phase gate. This means that it can be taken out of the universal set, but as the cubic phase gate is much more costly to implement, it is common to still include it. In general, Gaussian gates may be decomposed exactly in terms of squeezing, displacements and beamsplitters. We demonstrate how to achieve these decompositions in section 4.1.

Throughout this manuscript we express an arbitrary unitary operator as $U=e^{itH}$ where the Hermitian operator $H=\sum_{j=1}^{N}H_{j}$ is the generator of the gate. When decomposing arbitrary gates into a universal set, it is often necessary to approximate this sum of operators in the exponent as a product of exponential operators. More specifically, for $H=\hat{A}+\hat{B}$ where $\hat{A}$ and $\hat{B}$ are Hermitian operators, the Zassenhaus formula [43] states that

When $[\hat{A},\hat{B}]=0$ the product ends immediately after the first two operations. In general, however, it is possible that this product never terminates, resulting in a decomposition that is no longer finite. In this case, the product can be truncated at a designated stage in the expansion and the remaining commutators can be neglected. This strategy is referred to as a Trotter-Suzuki approximation [44, 45], which in general can be written as

where $H=\sum_{j=1}^{N}H_{j}$. This approximation requires $K=O(1/\varepsilon)$ gates to achieve precision $\varepsilon$ for fixed $t$. Note that the subscript $j$ on $H_{j}$ is only an index and does not refer to a mode, as each $H_{j}$ may contain any number of modes. In general, the Trotter-Suzuki approximation can be useful in combination with a variety of other decomposition techniques and approximations as we show in the next section.

In this section we examine an approximate decomposition technique detailed in Ref. [13]. This method expresses sums and products of the quadrature operators in terms of commutators and then approximates the exponentials of these commutators as repeated products of their arguments. More specifically, given a general unitary $U=e^{it\hat{H}}$ where $\hat{H}$ is a sum of products of quadrature operators, each term in the sum can be represented in terms of commutators with the following identities:

and for two-mode terms

Once the exponent of the general unitary operator is written in terms of a sum of commutators, it can be separated with the Trotter-Suzuki technique as in Eq. (8), and then expanded in the following way: given two Hermitian operators $\hat{A}$ and $\hat{B}$, it holds that [46]

Note that for fixed $t$, $K=O(1/\varepsilon)$ gates are required to achieve an error of $\varepsilon$ in this approximation, but the resulting circuit will have a depth of $O(1/\varepsilon^{2})$. It is possible then that very large circuits may be required to achieve a desired precision. A similar expression can also be used in the case of a nested commutator in the form $e^{t^{3}[\hat{B},[\hat{B},\hat{A}]]}$, given by

To illustrate the use of this decomposition technique, consider the decomposition of the unitary $e^{it\left(\hat{x}^{2}\hat{p}+\hat{p}\hat{x}^{2}\right)}$. First, using the equality from Eq. (10) we have that

Next, using Eq. (12) with $\hat{A}=\hat{x}^{3}$ and $\hat{B}=\hat{p}^{2}$ leads to

where each of the resulting gates on the right-hand side are part of the universal set in Eq. (5) up to Fourier transforms. In order to obtain a precision of $O(1/K)$, this product of gates must be repeated $O(K^{2})$ times. For example, if $t=1$ and the desired precision of the decomposition is $10^{-3}$, then the product of gates on the right-hand side needs to be repeated approximately $10^{5}$ times.

Another operator we can decompose to demonstrate this technique is the Kerr operation. We also examine the decomposition of this operation with a different method in section 5. The Kerr operation in terms of quadrature operators can be written as $e^{it\left(\hat{x}^{4}+\hat{x}^{2}\hat{p}^{2}+\hat{p}^{2}\hat{x}^{2}+\hat{p}^{4}-\hbar\hat{x}^{2}-\hbar\hat{p}^{2}\right)}$. After splitting the terms in the exponent using Trotter-Suzuki as in Eq. (8), and neglecting Fourier transforms, we are left to decompose operators which look like $e^{it\hat{x}^{4}}$ and $e^{it(\hat{x}^{2}\hat{p}^{2}+\hat{p}^{2}\hat{x}^{2})}$. These operators can be written in terms of commutators with Eq. (9) and (10) respectively to get

and

Expanding with Eq. (12) and (13), we get

and

where the second and fourth operators on the right-hand side of Eq. (18) can be expanded further, similar to the expansion in Eq. (15). Once again the product of gates on the right-hand sides of each equation must be repeated $O(K^{2})$ times to obtain a precision of $O(1/K)$. In this case, because the original operator needs to be expanded first with a Trotter-Suzuki approximation, the final sequence of gates needs to again be repeated to increase the precision as in Eq. (8). For derivations of some of these identities, as well as discussion on the necessity of good approximations, see the supplementary materials section of [13]

To potentially avoid the cost of repeating gates for greater precision, exact decomposition methods can be used as we show in the next section.