Basic circuit compilation techniques for an ion-trap quantum machine

The controlling apparatus allows the application of the single-qubit rotation $R(\theta ,\phi )$ described by the following unitary evolution operator:

$$\begin{eqnarray*}R(\theta ,\phi ):= \left(\begin{array}{rr}\cos \tfrac{\theta }{2} & -{{\rm{i}}{\rm{e}}}^{-{\rm{i}}\phi }\sin \tfrac{\theta }{2}\\ -{{\rm{i}}{\rm{e}}}^{{\rm{i}}\phi }\sin \tfrac{\theta }{2} & \cos \tfrac{\theta }{2}\end{array}\right).\end{eqnarray*}$$

Both θ and ϕ can be controlled by changing the duration and phase of the Raman beatnote that drives the Rabi oscillation of the qubit [6].

2.1.1. Single-qubit gate cost

The gate $R(\theta ,\phi )$ has two cost parameters,

$$\begin{eqnarray}&&d:=\displaystyle \frac{| \theta | }{\pi }{\tau }_{1q},\,{\rm{a}}{\rm{n}}{\rm{d}}\,e={e}_{1}:=| \sin (\theta )| \varepsilon \,{\rm{o}}{\rm{r}}\,e={e}_{2}:=(| \theta | {\rm{m}}{\rm{o}}{\rm{d}}\pi )\varepsilon ,\end{eqnarray} \tag{ 1 }$$

where d is the duration of the above single-qubit rotation and e gives a model of the experimental error based on laser pulse area fluctuations due to laser intensity and timing jitter, leading to random over-/under-rotations of the qubit. The formula describing e₁ is constructed such as to highlight the effect of the slope of the Rabi oscillation being smallest for full π rotations when the gate is applied to a quantum state close to the computational basis states [9]. For an unknown qubit state, it is impossible to tell the slope of Rabi oscillation, and thereby e₁ error model becomes inaccurate. The formula describing e₂ is designed such as to highlight that the error should be proportional to the rotation angle [10]. However, e₂ has its own limitations. Indeed, such a definition predicts that, to consider a specific example, the error in the single pulse circuit $R(\pi /2,0)$ will be smaller than that in the gate $R(\pi ,0)$ (both applied to a computational basis state). However, the experiment shows the opposite result—$R(\pi ,0)$ is more accurate than $R(\pi /2,0)$, and thereby e₂ is also inaccurate.

We note that while proper explanation and accurate modeling of experimental errors is very important, narrowing down a complete error model is not the focus of this paper. For the purpose of this paper, any error model can be acceptable. This is because the goal is to illustrate that trapped ions experiments can be optimized across a combination of (two) conflicting optimization criteria by those techniques reported, and to highlight the inner workings of such optimization approach. With this in mind, we select e₁ model, since optimization over e₂ is equivalent to optimization of the duration, reducing the number of optimization criteria to just one.

Presently, single-qubit rotations, as well as the two-qubit gates are implemented serially. As a result, the overall runtime of a computational experiment, as described by its circuit, equals to the sum of the runtimes of the individual gates. Depending on the desired properties of the circuit, one may choose to optimize the overall runtime, the overall error, the overall number of gates (including keeping separate counts of the single-qubit and two-qubit gates), as well as any combined figure of the above. In this paper, we will describe the overall cost of an implementation as a length-2 vector (d, e), with the components corresponding to the overall duration and the overall error. The error component itself is described by the list (written as a linear combination) of all errors from all gates participating in the respective circuit, per the error model introduced for the individual gates. This definition of the error does not correspond to the actual error seen in the experiment, but rather shows the influence and sources or errors within the given implementation. We try to minimize the cost vector (d, e), focusing separately on the duration and error. One may choose to focus on other optimization criteria, such as, e.g., minimization of the gate count; this does not affect the overall optimization strategy or the steps taken to arrive at the optimized solution.

For future discussions, we will need the following relations:

• ${R}^{-1}(\theta ,\phi )=R(\theta ,\phi -\pi )$, that can be used to construct the inverse of the $R(\theta ,\phi )$ gate at the same cost as the original gate; and
• $R(\theta ,\phi )=(-1)\cdot R(\theta -2\pi ,\phi )$, that can be helpful in that it provides the means for limiting the duration of any one $R(\theta ,\phi )$ gate to at most ${\tau }_{1q}$, as the global phase does not matter.

Both identities are easy to verify directly.

The single-qubit rotations around the basis axes must be expressed in terms of the physical-level R gate to be implementable in an experiment.

RX: Setting $\phi =0$ in $R(\theta ,\phi )$ achieves the rotation about the X axis by the angle θ, as follows:

$$\begin{eqnarray}{RX}(\theta ):= \left(\begin{array}{rr}\cos \tfrac{\theta }{2} & -{\rm{i}}\sin \tfrac{\theta }{2}\\ -{\rm{i}}\sin \tfrac{\theta }{2} & \cos \tfrac{\theta }{2}\end{array}\right)=R(\theta ,0).\end{eqnarray} \tag{ 2 }$$

Observe that the duration of ${RX}(\theta )$ is $\tfrac{| \theta | {\tau }_{1q}}{\pi }$, whereas its error is $| \sin \theta | \epsilon$, i.e., its cost vector is $\left(\tfrac{| \theta | {\tau }_{1q}}{\pi },| \sin \theta | \epsilon \right)$. The implementation (2) is well known.

RY: Setting $\phi =\tfrac{\pi }{2}$ in $R(\theta ,\phi )$ obtains the rotation about Y axis by the angle θ. In particular,

$$\begin{eqnarray}{RY}(\theta )\,:=\,\left(\begin{array}{rr}\cos \tfrac{\theta }{2} & -\sin \tfrac{\theta }{2}\\ \sin \tfrac{\theta }{2} & \cos \tfrac{\theta }{2}\end{array}\right)=R(\theta ,\pi /2).\end{eqnarray} \tag{ 3 }$$

As a result, the cost of ${RY}(\theta )$ is $\left(\tfrac{| \theta | {\tau }_{1q}}{\pi },| \sin \theta | \epsilon \right)$. The implementation (3) is also well known. The costs of the RX and RY gates with the same rotation angle are thus the same. RZ is more difficult to obtain. In particular,

RZ: RZ rotation is defined as follows,

$$\begin{eqnarray*}{RZ}(\theta ):=\left(\begin{array}{cc}{{\rm{e}}}^{-{\rm{i}}\theta /2} & 0\\ 0 & {{\rm{e}}}^{{\rm{i}}\theta /2}\end{array}\right).\end{eqnarray*}$$

It is easy to show that it cannot be obtained via a single physical R pulse, and thus requires a circuit with two or more R gates. Firstly, we found the following circuit implementing the RZ gate

$$\begin{eqnarray}{RZ}(\theta ) & = & {RY}\left(-v\displaystyle \frac{\pi }{2}\right).{RX}(v\theta ).{RY}\left(v\displaystyle \frac{\pi }{2}\right)\\ & = & R\left(-v\displaystyle \frac{\pi }{2},\displaystyle \frac{\pi }{2}\right).R(v\theta ,0).R\left(v\displaystyle \frac{\pi }{2},\displaystyle \frac{\pi }{2}\right),\end{eqnarray} \tag{ 4 }$$

where $v\in \{-1,+1\}$ is a variable allowing to arbitrarily set the sign of either first or last RY rotation, and '.' denotes matrix multiplication (recall that the order of gates in the circuit is given by the inverted order of matrices in the matrix product). The implementation with v = 1 was known to [6]; as we will show later, the ability to choose v, being our contribution to the above circuit, is very important in circuit optimization. The cost of this implementation is $\left(\tfrac{| \theta | {\tau }_{1q}}{\pi }+{\tau }_{1q},2\times \epsilon +1\times | \sin \theta | \epsilon \right)$. Alternatively, ${RZ}(\theta )$ gate may be obtained as

$$\begin{eqnarray}&&{RZ}(\theta )\equiv R(\pi ,x).R(\pi ,x-\theta /2),\end{eqnarray} \tag{ 5 }$$

up to an undetectable global phase of −1 (equality up to a global phase is furthermore denoted by '$\equiv$'), where the parameter x may be set arbitrarily. The cost of this implementation is $(2{\tau }_{1q},2\times | \sin \pi | \epsilon )=(2{\tau }_{1q},0)$. Observe that with the slightly longer execution time this second realization is associated with a smaller error, which seems to be a preferred scenario in the physical experiments if the gate is to be implemented by itself (as opposed to as a part of a larger computation). The flexibility in setting x within the above implementation allows to optimize quantum circuits where RZ is one of the gates used, as varying the value x allows to obtain either the ${RX}(\pi )$ gate or the ${RY}(\pi )$ gate to be either first or last gate in (5), and those may cancel out with other gates in the circuit. Varying parameter x furthermore allows optimizing classical control, as selecting a value of the R gate parameter used to implement a previous single-qubit gate allows to keep the phase of the Raman beatnote used a constant. This, however, is only a minor improvement to the classical control sequences. Implementation (4) may become more desirable for the purpose of circuit optimization, since it relies on the efficiently optimizable sequence of RX and RY gates.

The RZ gate may be implemented directly without resorting to a circuit-level composition of pulses by individually addressing the qubits with laser beams that result in a qubit energy level shift through the Stark effect [11, 12]. Physical-level RZ gate gives an advantage over the physical-level R gate when one desires to construct the ${RZ}(\theta )$, as the ${RZ}(\theta )$ requires two physical-level R gates, and only one physical-level RZ gate to be implemented. However, when such ${RZ}(\theta )$ is an internal gate to the circuit (and most gates in interesting quantum computations are internal), it can be written as either $R(\pi ,0).R(\pi ,-\theta /2)$ or $R(\pi ,\theta /2).R(\pi ,0)$ (5), where $R(\pi ,0)={RX}(\pi )$ can be commuted past the two-qubit XX gate (discussed in section 3) selectably to either left or right. This results in the effective ability to implement an internal ${RZ}(\theta )$ gate with just a single physical-level R gate. We furthermore note that in our optimized implementations of those circuits we tried, whenever the goal is to have no more than two sequential internal R gates apply to a given qubit in a sequence, this was always possible to accomplish. While this is unlikely to scale to arbitrary quantum computations, it is perhaps true that in practical designs most frequently no more than one R gate is required between a pair of two-qubit XX gates acting on a given qubit. This illustrates the expressive power of the R gates when used in conjunction with the two-qubit XX gates. To conclude this discussion, we believe the ability to implement RZ directly may not be in high demand, unless the properties of such a physical-level RZ gate, including its duration and error, are superior to the R gate.

${RX}(\pi )$, ${RY}(\pi )$, and ${RZ}(\pi )$ implement Pauli-X, Pauli-Y, and Pauli-Z gates up to an undetectable global phase of $-{\rm{i}}$. ${RX}(\pi /2)$ implements the square-root-of-NOT gate $V:= \tfrac{1+{\rm{i}}}{2}\left(\begin{array}{cc}1 & -{\rm{i}}\\ -{\rm{i}} & 1\end{array}\right)$ up to a global phase. The rotation ${RZ}(\pi /2)$ implements the quantum phase gate (commonly referred to as P or S) $P:= \left(\begin{array}{cc}1 & 0\\ 0 & {\rm{i}}\end{array}\right)$ up to a global phase. The quantum $\pi /8$ gate also known as the T gate, $T:=\,\left(\begin{array}{cc}1 & 0\\ 0 & \tfrac{1+{\rm{i}}}{\sqrt{2}}\end{array}\right)$, is obtained as ${RZ}(\pi /4)$, up to a global phase.

Recall that RX, RY, and RZ gates do not commute, but their parameters may be added, i.e., $G(a)G(b)=G(a+b)$, when G is either one of RX, RY, or RZ. This is important for the circuit optimization technique discussed later.

A common single-qubit gate that may not be expressed as an axial rotation with a certain parameter is the Hadamard gate, $H:= \tfrac{1}{\sqrt{2}}\left(\begin{array}{cc}1 & 1\\ 1 & -1\end{array}\right)$. It can be implemented up to a global phase as one of the following two circuits.

$$\begin{eqnarray}H & \equiv & {RY}(-\pi /2).{RX}(\pi )=R(\pi /2,\pi /2).R(\pi ,0),\\ H & \equiv & {RX}(-\pi ).{RY}(\pi /2)=R(-\pi ,0).R(\pi /2,\pi /2).\end{eqnarray} \tag{ 6 }$$

The cost of each of the above circuits is $(1.5{\tau }_{1q},\epsilon )$. As such, the Hadamard gate is, roughly speaking, as expensive as the RZ rotation, and more expensive than either RX or RY. The ability to choose which of the RX/RY gates in the decomposition of the Hadamard gate comes first and which is second is important to the optimization of quantum circuits.

An arbitrary single-qubit unitary gate can be written as a matrix $U={{\rm{e}}}^{{\rm{i}}{d}}\left(\begin{array}{rr}{{\rm{e}}}^{{\rm{i}}{a}}\cos b & {{\rm{e}}}^{{\rm{i}}{c}}\sin b\\ -{{\rm{e}}}^{-{\rm{i}}{c}}\sin b & {{\rm{e}}}^{-{\rm{i}}{a}}\cos b\end{array}\right)$ of four real-valued parameters $a,b,c,$ and d. We found the following implementation as a circuit with at most two physical-level R gates,

$$\begin{eqnarray}U\equiv \left(\begin{array}{rr}{{\rm{e}}}^{{\rm{i}}{a}}\cos b & {{\rm{e}}}^{{\rm{i}}{c}}\sin b\\ -{{\rm{e}}}^{-{\rm{i}}{c}}\sin b & {{\rm{e}}}^{-{\rm{i}}{a}}\cos b\end{array}\right)=R(-\pi ,-c-\pi /2).R(2b+\pi ,a-c-\pi /2).\end{eqnarray} \tag{ 7 }$$

The cost of this implementation is $\left({\tau }_{1q}+\tfrac{| 2b\mathrm{mod}\pi | {\tau }_{1q}}{\pi },| \sin (2b)| \epsilon \right)$. Observe, that equation (7) uses the minimal number of physical-level gates R required to implement an arbitrary single-qubit unitary. This can be established by counting the number of the real-valued degrees of freedom in the 2 × 2 unitary matrices up to global phase, and comparing it to the number of the real-valued degrees of freedom of the R gates. Equation (7) furthermore gives rise to the following lemma providing a guarantee on the cost of quantum physical-level circuits.

Lemma 1. Any quantum physical-level circuit over n qubits with $G$ two-qubit ${XX}$ gates can be reduced to an equivalent one with no more than $2(n+2G)$ single-qubit $R$ gates, providing, across all single-qubit gates used, an overall contribution of no more than $2{\tau }_{1q}(n+2G)$ to the runtime and a term of no more than $(n+2G)\times \epsilon$ to the error.

Proof. First, count the number of the 'pieces of wire', defined as an uninterrupted by any two-qubit gate qubit evolution time piece between two two-qubit gates in the circuit. This number is given by the expression $n+2G$, as is easy to verify by induction on G, the number of the two-qubit gates. Indeed, there are n pieces of wire in quantum circuits with no two-qubit gates, and the introduction of a two-qubit gate at the end of the circuit increases the number of the pieces of wire by two (specifically, on those qubits that the given two-qubit gate operates on). The single-qubit operations are contained to the individual pieces of wire, allowing to conclude the proof by referring to the equation (7) and the definitions of the duration and error.□

The above lemma reports an upper bound on the number of single-qubit gates, and could be used as a bottom line comparison for evaluating the efficiency of the optimizing compiler developed and tested in sections 4 and 5. We furthermore observe that the above lemma applies to show that the 5-qubit 80-gate QFT5 circuit, of which 10 are two-qubit gates reported in [6] is suboptimal. This is because according to lemma 1 the upper bound on the number of gates in such a circuit is 60.

The physical-level two-qubit gate available to us is the so-called ${XX}(\chi )$ gate, with parameter χ that depends on the pair of ions the gate is being applied to. The gate itself is defined by the following unitary matrix [6]:

$$\begin{eqnarray*}{XX}(\chi )=\left(\begin{array}{rrrr}\cos (\chi ) & 0 & 0 & -{\rm{i}}\sin (\chi )\\ 0 & \cos (\chi ) & -{\rm{i}}\sin (\chi ) & 0\\ 0 & -{\rm{i}}\sin (\chi ) & \cos (\chi ) & 0\\ -{\rm{i}}\sin (\chi ) & 0 & 0 & \cos (\chi )\end{array}\right).\end{eqnarray*}$$

The absolute value of the phase, $| \chi |$, can be set to an arbitrary real number between 0 and $\pi /2$ by varying the laser power used in the experiment [6]. The sign of χ depends on the laser detuning which is chosen based on the normal modes a particular pair of ions interacts with most strongly and hence which qubits the gate is being applied to [6]. The sign for each two-qubit gate is thus fixed experimentally and becomes an input parameter for how one is allowed to construct circuits.

The ${XX}(\chi )$ gate implements the well-known Mølmer–Sørensen gate [13], and latter is known to generate the CNOT gate (for $\chi =\pm \pi /4$) using single-qubit operations on the input and the output side. The CNOT gate is an important computational primitive—the ability to obtain it, coupled with the ability to implement any single-qubit gate, see equation (7), gives computational universality [14]. However, the ability to vary parameter χ to accept values beyond $\pm \pi /4$ allows a more efficient implementation of important quantum gates. Once computational universality is obtained, the efficiency becomes a next important step.

The following property of the ${XX}(\chi )$ gate is important to note for future discussions: ${XX}(\chi )$ commutes with any single-qubit ${RX}(\theta )$ rotation. However, ${XX}(\chi )$ does not commute with either ${RY}(\theta )$ or ${RZ}(\theta )$ when $\theta \not\equiv 0\mathrm{mod}2\pi$.

3.1.1. Cost function variables

The two-qubit ${XX}(\chi )$ gate has the vector-function cost (d, e), where [9]:

$$\begin{eqnarray}&&d:={\tau }_{2q},\ \mathrm{and}\ e={e}_{1}:=\,| \sin (2\chi )| E\ \mathrm{or}\ e={e}_{2}:= E.\end{eqnarray} \tag{ 8 }$$

Here, d is the duration of the two-qubit XX interaction, and e either models the error due to fluctuations in the experiment, analogous to the single-qubit case (e₁), or simply accepts a constant value E that is independent of the rotation angle (e₂). Current experiment [6] is setup such that the two-qubit XX gates can be applied to any pair of qubits in a serial fashion. Compared to the single-qubit gate R, the two-qubit gate is substantially longer in runtime and has a higher error. As a result, efficient circuit implementations must prefer the minimization of the use of the two-qubit XX gate over minimizing the single-qubit gates.

Depending on the sign s of the interaction χ for the given pair of qubits that we want to apply the CNOT gate to, it can be implemented up to the global phase of ${(-1)}^{-\tfrac{{vs}}{4}}$ such as shown in figure 1. Observe that v may be chosen arbitrarily from the set $\{-1,+1\}$ to set the signs of the rotation angle in RY at the beginning or at the end of the implementation. In particular, parameter v may be chosen such as to set the angle of the first RY rotation to the positive number, $+\tfrac{\pi }{2}$, in which case the second RY features the negative sign, or vice versa. The ability to choose the sign is particularly important in allowing single-qubit RY gate cancellations while decomposing logical-level circuits with multiple CNOT gates into efficient physical-level circuits.

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The cost of the above implementations of the CNOT gate is $(2{\tau }_{1q}+{\tau }_{2q},4\times \epsilon +1\times E)$.

Other than using two fewer single-qubit gate pulses compared to [15], our CNOT implementation allows to arbitrarily set the values of the RY rotations, and also allows further transformations and optimizations per template (12) discussed later. Our CNOT implementation saves two single-qubit pulses $\left(\mathrm{one}\,{RX}\left(\pm \tfrac{\pi }{2}\right)\right.$ and one $\left.{RY}\left(\pm \tfrac{\pi }{2}\right)\right)$ over the one reported in [6].

Controlled roots of axial rotations (Pauli gates) play an important role in quantum circuits. For instance, the n-qubit quantum Fourier transform is best viewed as a circuit with $\tfrac{n(n-1)}{2}$ controlled roots of Pauli-Z gates [1, figure 5.1]. The controlled-sqrt-NOT gate is used in the construction of an efficient five two-qubit gate circuit implementing the Toffoli gate [14, lemma 6.1]. Otherwise, if the proper root is not available, and the CNOT gate is the only two-qubit gate directly constructible, the Toffoli gate requires six two-qubit physical gates [16]. It is furthermore known that each controlled unitary gate can be implemented with the use of two CNOT gates, and, equivalently, two Mølmer–Sørensen gates, along with some single-qubit gates [14, lemma 5.1]. We next show that only one ${XX}(\chi )$ gate suffices to implement any controlled root of a Pauli gate, providing an improvement by a factor of two. In particular, depending on the sign s of χ, the controlled-${X}^{\alpha }$, $\alpha \in {\mathbb{R}},-1\leqslant \alpha \leqslant 1,$ may be obtained as follows:

The inverse of the controlled-${X}^{\alpha }$ may be obtained from the above circuit by attempting to construct the controlled-${X}^{-\alpha }$. Observe that the decomposition with the sign opposite to the physically available sign of χ needs to be selected. Other controlled roots of Paulis, such as the controlled-${Y}^{\alpha }$ and the controlled-${Z}^{\alpha }$ are related to the controlled roots of the NOT gate by the following formulas,

As a result, all controlled roots of Paulis are constructible using at most one physical-level two-qubit XX gate.

In our constructions, we favored the decompositions that feature ${RY}\left(\pm \tfrac{\pi }{2}\right)$ gates with arbitrarily selectable sign of the rotation parameter. This was done to allow the selection of the specific sign or the relations between signs to dictate the RY gate cancellations. In some cases, the results of such cancellations can be dramatic, as is illustrated in the following lemma.

Lemma 2. Circuits over arbitrary ${RZ}$ rotations (including Pauli-$Z$, Phase, and $T$ gates) and the controlled-$Z$ can be written as an efficient trapped ions physical-level implementation as follows:

• 1.  
  The layer of ${RY}$ gates, defined as the set of gates ${RY}\left({v}_{i}\tfrac{\pi }{2}\right)$ applied to every qubit $i$, where ${v}_{i}\in \{-1,+1\}$ can be selected arbitrarily.
• 2.  
  The layer of ${RX}$, where qubit $i$ experiences the application of ${RX}\left(t-({v}_{{x}_{1}}{s}_{{{ix}}_{1}}+{v}_{{x}_{2}}{s}_{{{ix}}_{2}}\,+\,\cdots \,+\,{v}_{{x}_{k}}{s}_{{{ix}}_{k}})\tfrac{\pi }{2}\right)$, where $t$ is the aggregate angle accomplished by the combination of ${RZ}$ rotations applied to this qubit, $k$ is the number of ${CZ}$ gates that apply to the qubit $i$, and ${s}_{{ix}1},{s}_{{{ix}}_{2}},\ldots ,{s}_{{{ix}}_{k}}$ are the signs of the respective interactions.
• 3.  
  The set of ${XX}$ gates, where each controlled-Z gate ${CZ}(a,b)$ in the original circuit is represented by ${{XX}}_{{ab}}\left({s}_{{ab}}\tfrac{\pi }{4}\right)$.
• 4.  
  The layer of ${RY}$ gates, with ${RY}\left(-{v}_{i}\tfrac{\pi }{2}\right)$ applied to the qubit $i$.

Proof. Construct the controlled-Z gate as follows:

Observe that per formulas (4), (11) and upon the decomposition of RZ and controlled-Z gates into ${RX}/{RY}/{XX}$ circuits each qubit in each RZ/CZ gate experiences the application of ${RY}\left(v\tfrac{\pi }{2}\right)$ in the beginning and ${RY}\left(-v\tfrac{\pi }{2}\right)$ in the end. Therefore, setting the value of the parameter v equal to the previously used v accomplishes the following: every two RY gates between any two RZ/CZ gates in the target circuit cancel out. However, there will be a layer of RY in the beginning of the circuit and a layer of RY at the end with the opposite signs, for each qubit that experiences an application of a gate. Next, notice that all internal gates (those except the outside RY layers) are RX and XX, and thereby they all commute. This means we need only one layer of RX, and due to the formulas (4), (11), the rotation angle is calculated such as stated in the lemma. Each ${CZ}(a,b)$ in the original circuit is furthermore represented by ${{XX}}_{{ab}}\left({s}_{{ab}}\tfrac{\pi }{4}\right)$, per formula (11).□

Observe that allowing the layers of RY gates in lemma 2 to share one sign across all RY used (v_i = v_j for every pair of qubits i and j) allows their implementation with a single global pulse. This capability is currently not supported by the existing hardware [6], but may potentially be implemented in the future.

Recall that a quantum template is a quantum circuit with n gates that evaluates to the identity, ${G}_{0}{G}_{1}...{G}_{n-1}={\rm{Id}}$ [17]. A template can be used to construct a number of circuit identities,

$$\begin{eqnarray*}&&{G}_{i}{G}_{i+1\mathrm{mod}n}...{G}_{i+k-1\mathrm{mod}n}={G}_{i-1\mathrm{mod}n}^{-1}{G}_{i-2\mathrm{mod}n}^{-1}...{G}_{i+k\mathrm{mod}n}^{-1},\end{eqnarray*}$$

for arbitrary i and k, $0\leqslant i,k\leqslant n$, which may, in turn, be used to optimize quantum circuits via matching gates on the left hand side in the above equation and replacing them with the gates on the right hand side. Here we report one such template that is particularly useful in our constructions. Specifically, for any $a,b\in {\mathbb{R}}$ there exist $c,d\in {\mathbb{R}}$ such that:

$$\begin{eqnarray*}&&\mathrm{where}\qquad \qquad c:=2\,\arccos \left(\cos a\cdot \cos \displaystyle \frac{b}{2}\right),\\ d\,:=\,\left\{\begin{array}{ll}\arcsin \left(\displaystyle \frac{\sin \tfrac{b}{2}}{\sqrt{1-{\cos }^{2}a\cdot {\cos }^{2}\tfrac{b}{2}}}\right) & {\rm{if}}\ a\gt 0\\ \pi -\arcsin \left(\displaystyle \frac{\sin \tfrac{b}{2}}{\sqrt{1-{\cos }^{2}a\cdot {\cos }^{2}\tfrac{b}{2}}}\right) & {\rm{if}}\ a\lt 0.\end{array}\right.\end{eqnarray*}$$

The above template may be used in multiple ways.

• Firstly, it allows the replacement of ${RX}(a){RY}(b){RX}(a)$ with $R(c,d)$. The latter circuit always has a smaller duration and a higher overall fidelity (smaller contribution to the overall error), see figures 2(a), (c).
• This template may be used to replace ${RX}(a){RY}(b)$ with $R(c,d){RX}(-a)$ and ${RY}(b){RX}(a)$ with ${RX}(-a)$ $R(c,d)$. This allows to trade off runtime for error, see figures 2(b), (d) for the illustration of the changes in the runtime and error. While the runtime always increases, it is sometimes possible to improve the fidelity. For instance, if ${RX}\left(\tfrac{\pi }{2}\right){RY}\left(-\tfrac{\pi }{2}\right)$ in the circuit in figure 1 were replaced with $R\left(\pi ,-\tfrac{\pi }{4}\right){RX}\left(-\tfrac{\pi }{2}\right)$ this would result in the cost vector change of that part of the computation from $({\tau }_{1q},2\times \epsilon )$ to $(1.5{\tau }_{1q},\epsilon )$. This constitutes an increase of the runtime by $0.5{\tau }_{1q}$ over the increase of the fidelity from ${(1-\epsilon )}^{2}$ to $1-\epsilon$. Since the above rule applies to any pair RX and RY, it allows substantial flexibility in exchanging runtime for error.

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In our approach to the optimization of quantum circuits we favor gate decompositions relying on RX, RY, and XX gates. The efficiency is evidenced through short elementary gate decompositions (Z and its roots, Hadamard, CNOT, controlled-roots of Paulis), and favorable in-circuit gate cancellations, such as illustrated in lemma 2. Template (12) enables the next set of optimizations. Specifically, given a circuit over RX, RY, and XX gates, the template (12) allows to 'commute' arbitrary RX to the right (or left) of every RY met via replacing ${RX}(a){RY}(b)$ with the properly defined $R(c,d){RX}(-a)$. Observe that such 'commutation' changes the sign of the parameter in RX, as such it is a special kind of commutation. Recalling that ${RX}(a){RX}(b)={RX}(a+b\mathrm{mod}2\pi )$ and that RX commutes with XX allows to 'commute' all RX to the end (or beginning) of the circuit via replacing RY gates with R gates. This reduces the number of RX gates to at most one per qubit. This result is furthermore summarized in the following Lemma.

Lemma 3. Any quantum physical-level circuit over $n$ qubits with ${G}_{{XX}}$ two-qubit ${XX}$ gates, ${G}_{{RY}}$ single-qubit ${RY}$ gates, and ${G}_{{RX}}$ single-qubit ${RX}$ gates can be reduced to an equivalent one with no more than ${G}_{{XX}}$ two-qubit ${XX}$ gates, ${G}_{{RY}}$ single-qubit $R$ gates, and at most $n$ single-qubit ${RX}$ gates.

To implement the Boolean multiplication on the trapped ions machine our algorithm takes the following steps.

• Steps 1–3 identify that the computation we wish to perform is given by the Toffoli gate, ${\rm{TOF}}[a,b;c]$, and step 4 finds the following 5 two-qubit gate circuit implementing the Toffoli gate [14] using CNOT and controlled-$\sqrt{{\rm{NOT}}}$ gates,
  
• Physical qubits from the set $\{1,2,3,4,5\}$ are mapped onto logical qubits: we choose 2, 4, and 5 (top to bottom), such as to rely on the high fidelity interactions [6, table 1]. The controlled roots-of-NOT gates do not feature neighboring controls, imposing no further conditions on the mapping.
• • Now that the signs of ${\chi }_{i,j}$ are known, mark the controls of the controlled-$V/{V}^{\dagger }$ gates with $+/-$ by the sign of the RY rotation that appears on it when decomposed into physical pulses using formulas (9),
    
    and then choose the CNOT gate decompositions (figure 1) such as to maximize RY cancellations,
    
    The above choice for the CNOT decompositions allows to cancel two pairs of ${RY}\left(\tfrac{\pi }{2}\right)/{RY}\left(-\tfrac{\pi }{2}\right)$ gates on the first qubit, as is evidenced by the 'meeting' plus and minus signs.
  • Perform gate substitutions and cancellations to obtain the circuit shown in figure 3(a).
  • The rest of the single-qubit optimization algorithm is based on the template (12), and works differently depending on the criteria for the remaining optimization. If the runtime is the goal, the algorithm tries to find triples of gates ${RX}-{RY}-{RX}$ that may be replaced with the R gate, such as to minimize the overall duration. This allows to construct the circuit pictured in figure 3(b). In case if minimizing the error is preferred, the algorithm 'commutes' all RX to the left. The result of this optimization may be found in figure 3(c). We conclude the optimization by layering the single-qubit gates sharing the same duration, by moving RX, whenever possible, such as to allow their sequential execution—this helps to optimize classical controlling sequences.

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Table 1.  Benchmark circuits. Errors shown per both definitions of the single- and two-qubit R/XX gate error models, as described in (1), (8).

Function	#q	1q/2qg	Time	Error
Grover{011,111}	4	29/10	2743 $\mu {\rm{s}}$	${e}_{1}=6\times 0.707107\epsilon +4\times 0.866025\epsilon +9\times \epsilon +6\times 0.707107E+4\times E$
				${e}_{2}=4\times \tfrac{\pi \epsilon }{4}+9\times \tfrac{\pi \epsilon }{2}+3\times \tfrac{2\pi \epsilon }{3}+3\times \tfrac{3\pi \epsilon }{4}+10\times \pi \epsilon +10\times E$
Grover{011,101}	4	31/12	3250 $\mu {\rm{s}}$	${e}_{1}=6\times 0.707107\epsilon +3\times 0.866025\epsilon +10\times \epsilon +6\times 0.707107E+6\times E$
				${e}_{2}=4\times \tfrac{\pi \epsilon }{4}+10\times \tfrac{\pi \epsilon }{2}+3\times \tfrac{2\pi \epsilon }{3}+2\times \tfrac{3\pi \epsilon }{4}+12\times \pi \epsilon +12\times E$
Grover{010,100}	4	32/12	3280 $\mu {\rm{s}}$	${e}_{1}=6\times 0.707107\epsilon +3\times 0.866025\epsilon +10\times \epsilon +6\times 0.707107E+6\times E$
				${e}_{2}=3\times \tfrac{\pi \epsilon }{4}+10\times \tfrac{\pi \epsilon }{2}+3\times \tfrac{2\pi \epsilon }{3}+3\times \tfrac{3\pi \epsilon }{4}+13\times \pi \epsilon +12\times E$
Grover{000,111}	4	31/13	3492 $\mu {\rm{s}}$	${e}_{1}=4\times 0.707107\epsilon +5\times 0.866025\epsilon +10\times \epsilon +6\times 0.707107E+7\times E$
				${e}_{2}=3\times \tfrac{\pi \epsilon }{4}+10\times \tfrac{\pi \epsilon }{2}+5\times \tfrac{2\pi \epsilon }{3}+1\times \tfrac{3\pi \epsilon }{4}+12\times \pi \epsilon +13\times E$
QFT4	4	13/6	1582 $\mu {\rm{s}}$	${e}_{1}=1\times 0.273262\epsilon +1\times 0.382683\epsilon +2\times 0.521005\epsilon +1\times 0.866025\epsilon +1\times 0.92388\epsilon$
				$+\,2\times 0.980785\epsilon +4\times \epsilon +1\times 0.19509E+2\times 0.382683E+3\times 0.707107E$
				${e}_{2}=1\times \tfrac{3\pi \epsilon }{8}+4\times \tfrac{\pi \epsilon }{2}+2\times \tfrac{9\pi \epsilon }{16}+1\times \tfrac{2\pi \epsilon }{3}+2\times 0.825557\pi \epsilon +1\times \tfrac{7\pi \epsilon }{8}$
				$+\,1\times 0.911896\pi \epsilon +1\times \pi \epsilon +6\times E$
QFT5	5	22/10	2669 $\mu {\rm{s}}$	${e}_{1}=1\times 0.138284\epsilon +1\times 0.273262\epsilon +1\times 0.521005\epsilon +1\times 0.55557\epsilon +1\times 0.707107e$
				$+\,1\times 0.722528\epsilon +1\times 0.831147\epsilon +2\times 0.866025\epsilon +1\times 0.951173\epsilon +2\times 0.995185\epsilon$
				$+\,4\times \epsilon +1\times 0.098017E+2\times 0.19509E+3\times 0.382683E+4\times 0.707107E$
				${e}_{2}=1\times \tfrac{3\pi \epsilon }{16}+2\times \tfrac{15\pi \epsilon }{32}+4\times \tfrac{\pi \epsilon }{2}+1\times 0.59988\pi \epsilon +2\times \tfrac{2\pi \epsilon }{3}+1\times \tfrac{11\pi \epsilon }{16}$
				$+\,1\times 0.74298\pi \epsilon +1\times \tfrac{3\pi \epsilon }{4}+1\times 0.825557\pi \epsilon +1\times 0.911896\pi \epsilon$
				$+\,1\times 0.955841\pi \epsilon +6\times \pi \epsilon +10\times E$
Toffoli-4	5	21/11	2832 $\mu {\rm{s}}$	${e}_{1}=8\times 0.707107\epsilon +2\times 0.866025\epsilon +4\times \epsilon +3\times 0.707107E+8\times E$
				${e}_{2}=8\times \tfrac{\pi \epsilon }{4}+4\times \tfrac{\pi \epsilon }{2}+2\times \tfrac{2\pi \epsilon }{3}+7\times \pi \epsilon +11\times E$

Observe that at the stage of the decomposition of the two-qubit logical gates into implementable physical-level gates, the single-qubit pulse count was 20. The parametrized CNOT implementation per figure 1, along with the algorithm for joint gate decomposition, and RX gates commutation over the application of template (12) allowed to reduce the single-qubit pulse count from the original 20 down to 9 (figure 3(c)). Had the previously known circuitry implementing the CNOT gate with two more single-qubit pulses [6, 15] been used, the original unoptimized single-qubit gate count would have been 30. The upper bound on the number of single-qubit gates in a circuit of this size, as given by the application of lemma 1, is 26. Had the efficient controlled-root-of-Pauli implementation introduced in this paper been not used, the resource count would have been substantially higher—not only the single-qubit gates, but this time, two-qubit gates as well [16]. This illustrates the power of the approach reported in this paper.

Our Toffoli gate implementation, figure 3(b), takes time 1285 $\mu {\rm{s}}$, which compares favorably to 1500 $\mu {\rm{s}}$ reported in [24]. The advantage in the duration, however, is attributed to the different hardware [6] that we rely on in our work. The difference in the fidelity between our implementation and the one reported in [24] needs to be established via the experiment. The major difference between our implementation and the one reported in [24] is our circuit is a true Toffoli gate that can be used (and in fact is used in the next subsection) as a primitive in the implementation of quantum algorithms, whereas [24] reports a Toffoli gate implemented up to a relative phase; such an implementation up to the relative phase may not be used in quantum algorithms directly—specifically, it would give an incorrect answer if used within Grover's search [25].

Table 1 reports the result of the application of the above techniques to the design and optimization of circuits implementing advanced quantum computational experiments. None of the implementations proposed in table 1 have yet been demonstrated in an experiment. Table 1 lists the name of the algorithm/function, the number of qubits the developed implementation uses, the number of physical-level single-qubit/two-qubit pulses, the overall runtime of the circuit, and all sources of errors. All implementations reported in table 1 are true implementations of the respective algorithms/functions, in that we report exact unitaries (VS those up to a relative phase), treat black boxes as black boxes (no optimizations crossing black box boundaries), as well as precisely follow the formulation of the respective algorithms and specifications. Our computations are thus properly scalable to accept larger numbers of qubits. The quantum state is furthermore initialized to $| 00000\rangle$ (as opposed to a state containing partial results of a computation) before any of the circuits are applied.

For Grover's algorithm, we considered the scenario when the three-bit Boolean function $f(x)=f({x}_{1},{x}_{2},{x}_{3})$ implemented as Grover's oracle, $| x,y\rangle \mapsto | x,y\oplus f(x)\rangle$, marks some two items in the database. There are 28 such functions. The superscript in the name of the Grover's function in table 1 indicates the bit strings encoded by the oracle the search over which is being reported. The efficiency of the Grover's circuit is determined by the efficiency of the implementation of the oracle, which in turn is determined by the Hamming distance between those items it marks. We treat the oracle as a black box, and do not allow optimizations across the boundary of the black box. While we can implement Grover's algorithm over any oracle marking two items, only a few representative circuits are actually included in the table. To implement Grover's algorithm over a Boolean function marking one item, the oracle needs to be a Toffoli-4 gate (the triply controlled Toffoli, $\mathrm{TOF}[a,b,c;d]$). We calculated that the single Grover's iteration requires 16 two-qubit gates. The algorithmic probability of reading out the correct answer upon applying a single Grover's iterate is 0.78125, which beats the classical probability of finding the answer with the single query, 0.125, therefore, it may be interesting to run such an experiment, as well.

Grover's algorithm may furthermore be implemented using the oracle function g computing the unknown Boolean function f into phase, $g:| x\rangle \mapsto {(-1)}^{f(x)}| x\rangle$ [1]. For the 2-out-of-8 search the function g can be thought of as a Z/CZ circuit—see lemma 2 outlining the construction of efficient Z/CZ circuits. The number of CZ gates used ranges between one and three. This means that the entire Grover's search with such a phase oracle can be implemented using between 6 and 8 two-qubit XX gates. A 1-out-of-8 search with the single Grover's iterate and algorithmic success probability of 0.78125 over phase oracle can be accomplished with 10 two-qubit XX gates. Finally, some 4-qubit Grover's searches may be possible to demonstrate. Specifically, the search for phase items $\{1110,1111\}$ can be accomplished by a trapped ions circuit with 16 two-qubit XX gates over algorithmic success probability of 0.78125.

The QFT5 circuit reported in table 1 can be compared head-to-head to the one found in [6]. Specifically, both circuits are true [1, figure 5.1] (as opposed to semiclassical, [26]) implementations of the 5-qubit QFT, featuring 10 two-qubit gates. Our circuit benefited from careful design, and as a result features the single-qubit gate count of just 22 compared to 70 in [6]. This constitutes the reduction of the single-qubit gate count by a factor of more than three, clearly illustrating the benefits of our approach.

To our knowledge, the Toffoli-4 gate circuit reported in table 1 is the first such containing no more than 11 two-qubit gates. Previous best result is 12 CNOTs [20].