Limitations of Quantum Measurements and Operations of Scattering Type under the Energy Conservation Law

It is essential to improve the accuracy of quantum measurements and quantum gate operations in quantum information processing. Quantum computing attracts industrial attention, since Shor [1, 2] found an efficient quantum algorithm for prime factorization, the hardness of which is assumed in some protocols for public key cryptography. However, it is demanding to reduce the error in gate operations and measurements below the rate given by the threshold theorem for successful error correction to realize fault-torelant quantum computation [3, 4, 5, 6, 7]. Quantum key distribution [8], which aims to realize an information-theoretically secure cryptosystem, also demands to reduce the error to perform the protocol effectively, since any error, even not caused by eavesdroppers, reduces the key rate by error reconciliation. Other examples that require high accuracy of measurements in fundamental physics are the observation of the polarization of photons in the cosmic background radiation, which is thought to carry information about the early universe [9, 10], and the observation of gravitational waves, which has recently been used as a tool to elucidate the origin of elements and the early universe [11].

On the other hand, it is known that a conservation law limits the accuracy of measurements and unitary operations. Wigner [12] first showed in 1952 that the projective measurement of the spin $x$-component of a particle with spin $\frac{1}{2}$ cannot be realized under the conservation of angular momentum along the $z$-axis. Later, Araki and Yanase [13] showed the no-go theorem which states that, in general, an additive conservation law limits the accuracy of projective measurement of the quantity not commuting with the conserved quantity. This theorem is called the Wigner–Araki–Yanase theorem (WAY theorem).

Ozawa [14, 15, 16] obtained quantitative generalizations of the WAY theorem by introducing a systematic method of manipulating the noise commutation relations, the commutation relations between the noise operator, previously introduced by von Neumann [17, p. 404], and the conserved quantity, and established the tradeoff between the measurement error and the fluctuation of the conserved quantity in the apparatus for arbitrary measurements, originally suggested by Yanase [18] for spin measurements. Ozawa [19] also derived a universally valid reformulation of Heisenberg’s error-disturbance relation and in [20] showed that the quantitative generalization [16] of the WAY theorem is a consequence of the conservation law and Heisenberg’s error-disturbance relation in its revised form.

The noise commutation relations were also used to evaluate the accuracy of gate implementations under conservation laws for the CNOT gate[21], followed by quantitative evaluations of the gate fidelity for the Hadamard gate and the NOT gate [20, 22], showing that to be the smaller the error, it is necessary to make the greater the fluctuation of the conserved quantity in the controller, or the size of the controller. The above approach was compared with Gea-Banacloche’s [23, 24, 25, 26] model dependent approach to the error in quantum logic gates caused by the quantum nature of controlling systems [27, 28]. Karasawa and his collaborators [22, 29] proved the implementation error bound for the NOT gate and later for arbitrary one-qubit gates. Afterwards, Tajima et al. [30, 31] provided a tighter lower bound for the error of quantum measurement, and implementations of quantum gates under conservation laws by expressing the fluctuation of the conserved quantity in terms of the quantum Fisher information.

The WAY theorem tells us that when we want to apply operations that break the symmetry, we must compensate the system’s conserved quantity fluctuations, i.e., the coherence with respect to conserved quantities, as a resource. This point of view has been systematized as the quantum resource theory of asymmetry, and the operations that are feasible in the presence of symmetry have been investigated [32, 33, 34, 35, 36, 37, 38, 39, 40]. The WAY theorem has also been applied to the black hole information loss problem [41, 42, 43] to investigate to which degree of completeness the scrambled information can be recovered in the presence of symmetry [44, 45].

Although most of the work on the WAY theorem has been done on additive conservation laws that do not include interaction terms, such as momentum and angular momentum conservation laws, some researchers extended the WAY theorem to multiplicative conservation laws [46] and non-additive conservation laws such as energy conservation laws because it contains interaction terms [47, 48, 49]. Navascués et al proved relations between the difference between a desired POVM measurement we want to implement and an actual measurement we are able to implement under the energy conservation and the energy distribution[47]. Miyadera et al. showed an inequality between the fidelity of information distribution and the norm of the interaction term of Hamiltonian[48]. Tukiainen proved a WAY-type no-go theorem beyond conservation laws in the assumption of the weak Yanase condition[49]. We are able to obtain an upper bound of gate fidelity by the WAY theorem in general, it is also known that there is a lower bound of gate fidelity under the energy conservation law using methods in the resource theory [50].

In this paper, we prove that the relation between a measurement error, non-commutativity of the measured quantity and the conserved quantity, and the variance of the conserved quantity under an additive conservation [16] is able to be extended to measurements in scattering processes under energy conservation and we have a similar lower bound of the measurement error. We also show an upper bound of the gate fidelity of SWAP gate as an application of the theorem. Furthermore, we prove necessary conditions for the perfect implementation of a controlled unitary gate in a scattering process under energy conservation. In addition, we also show an inequality between the gate fidelity and energy variances for two-qubits controlled unitary gate, which is the extension of Ozawa’s result [21].

One of the possible applications of our results for fundamental science is a design of gravitational wave detectors. Recently some analysis of the dynamics of the gravitational wave as a quantum field are known [51, 52, 53]. In the observation of a gravitational wave using a laser interferometer, the information of the gravitational wave is transfered to the laser interferometer via the gravitational interaction. Then we are able to know the state of the gravitational wave by measuring the state of the laser interferometer. However the energy conservation gives the constraints for measurement, we have to design detectors to lower the measurement error.

Section 2 presents a lower bound for the measurement error of quantum measurements in scattering processes, which are important in particle physics experiments, under the requirement of energy conservation. In section 3, we show the conditions that the free Hamiltonian of the control system must satisfy in order to obtain a vanishing implementation error of the control unitary gates in the case of energy conservation. We also give an inequality between the energy fluctuation and the gate fidelity of two-qubit controlled unitary gates under energy conservation in section 3. Finally, we summarize the results in section 4.

In this section, we present an error limit of quantum measurements of scattering type under the energy conservation law.

Firstly, we review Ozawa’s inequality [19] and the quantification of the WAY theorem using it [16, 20] for our later discussion. Let’s consider that we want to measure an observable $A_{S}$ of a system $S$ indirectly using a meter observable $M_{D}$ of a detector system $D$, in other words, we measure $A_{S}$ at the initial time $t=0$ by measuring the meter observable $M_{D}$ at $t=\tau$. In the Heisenberg picture, we indirectly measure $A_{S}(0)$ by measuring $M_{D}(\tau)=U^{\dagger}M_{D}U$, where $U$ is a unitary time evolution operator from time $t=0$ to $t=\tau$ that describes the process in which $S$ and $D$ become correlated. The output probability distribution of this indirect measurement is described by a POVM as follows. Let us denote a spectral decomposition of $M_{D}$ by $M_{D}=\sum_{m}mP_{m,D}$, where $P_{m,D}$ is a projection operator associated with an eigenvalue $m$ of $M_{D}$. Let us suppose that $S$ and $D$ are not correlated in the initial time and the initial state of the composite system $S$+$D$ is given by $\rho_{S}\otimes\sigma_{D}$. The probability that the measured value of $M_{D}$ is $m$, represented as ${\rm Pr}(M_{D}=m||\rho_{S})$ , is given by ${\rm Pr}(M_{D}=m||\rho_{S})={\rm Tr}_{SD}[(I_{S}\otimes P_{m,D})U(\rho_{S}\otimes\sigma_{D})U^{\dagger}]$. Defining a POVM element $E_{m}$ by $E_{m}={\rm Tr}_{D}[(I_{S}\otimes\sigma_{D})U^{\dagger}(I_{S}\otimes P_{m,D})U]$, ${\rm Pr}(M_{D}=m||\rho_{S})={\rm Tr}_{S}[\rho_{S}E_{m}]$ holds.

We define the error operator ${\bf E}$ for the measurement of $A_{S}$ and the disturbance operator ${\bf D}$ for an observable $B_{S}$ as follows:

In the indirect measurement, we indirectly measure $A_{S}(0)=A_{S}\otimes I_{D}$ by measuring $M_{D}(\tau)$ $=U^{\dagger}(I_{S}\otimes M_{D})U$, we defined the error operator by the difference between $A_{S}(0)$ and $M_{D}(\tau)$. Similarly, we defined the disturbance operator by the difference between $B_{S}$ at $t=0$ and $B_{S}$ at $t=\tau$. We define the measurement error $\varepsilon(A_{S})$ and the disturbance $\eta(B_{S})$ by $\varepsilon(A_{S})=\sqrt{{\rm Tr}[{\bf E}^{2}(\rho_{S}\otimes\sigma_{D})]}$ and $\eta(B_{S})=\sqrt{{\rm Tr}[{\bf D}^{2}(\rho_{S}\otimes\sigma_{D})]}$. Then the following Ozawa’s inequality

holds[19], where $\sigma(A_{S})$ and $\sigma(B_{S})$ represent standard deviations of $A_{S}$ and $B_{S}$ in the state $\rho_{S}$, respectively.

Next, we review the lower bound of the measurement error $\varepsilon(A_{S})$ when there is an additive conserved quantity [16]. We suppose that $L=L_{S}\otimes I_{D}+I_{S}\otimes L_{D}$ is an additive conserved observable for the measurement process. For example, the angular momentum and the momentum are additive conserved observables. When we have an additive conserved quantity $L$, the time evolution operator $U$ which characterizes the measurement process must satisfy the following condition:

We further assume that the commutativity of the meter observable $M_{D}$ and the detector’s conserved quantity $L_{D}$, which is called Yanase’s condition, i.e., $[M_{D},L_{D}]=0$. Then it is known that

holds [16]. In [20] it is shown that Eq. (5) is a consequence of Ozawa’s inequality [19]. This enables us to evaluate the lower bound of the measurement error quantitatively when there is an additive conserved observable.

In the following, we show that Eq. (5) is able to be extended to measurements in scattering processes under energy conservation and we have a similar lower bound of the measurement error. In the following, we consider two spin $\frac{1}{2}$ particles I and II. Each of them has one degree of freedom in position space (the orbital degree) and the spin degree of freedom. We represent by $O_{\rm I}$ and $O_{\rm II}$ the orbital degrees of freedom of I and II, respectively. We denote by $S$ and $D$ the spin degrees of freedom of I and II, respectively. The observable $A_{S}$ is assumed to be an operator on the spin state space of I. Let I and II are initially spatially separated and having not interacted before. Then, as I approaches II, they start to interact, creating a correlation between I and II. Then, as I moves away from II, after a certain time the interaction stops. We can indirectly measure the system $S$ by measuring the system $D$, after it is correlated with the system $S$.

Let $H_{{\rm I+II}}$ be the time-independent Hamiltonian of the composite system I + II, given by

where $H_{\rm I}$ is the Hamiltonian of I, $H_{\rm II}$ is the Hamiltonian of ${\rm II}$ and $H_{int}$ is the interaction term between ${\rm I}$ and ${\rm II}$. We consider particular initial states in which the interaction term is not effective in the time period $(0,\tau)$. The time evolution operator of the scattering process $U_{{\rm I}+{\rm II}}$ is defined by

where we employed natural units, $\hbar=1$.

Let us suppose that the initial state of ${\rm I}$ is $\ket{\psi}\ket{\chi}$ and that of ${\rm II}$ is $\ket{\phi}\ket{\xi}$, where $\ket{\psi}$ and $\ket{\phi}$ are wavefunctions of the orbital degrees of ${\rm I}$ and ${\rm II}$, respectively, and moreover $\ket{\chi}$ and $\ket{\xi}$ are state vectors of $S$ and $D$, respectively. The final state of the composite system ${\rm I}$+${\rm II}$ at $t=\tau$ is $U_{{\rm I+II}}(\ket{\psi}\ket{\chi}\otimes\ket{\phi}\ket{\xi})$. We assume that particles ${\rm I}$ and ${\rm II}$ are spatially localized and separated at $t=0$, and the interaction $H_{int}$ is a short range interaction. Formally, we assume the following condition on the initial states and $H_{\rm I+II}$:

Note that we consider some specific initial orbital states $\ket{\psi}$ and $\ket{\phi}$ which satisfy Eq. (8). We do not impose Eq. (8) for all orbital states of ${\rm I}$ and ${\rm II}$. On the other hand, we consider arbitrary spin states $\ket{\chi}$ for $S$ and an arbitrary but fixed state $\ket{\xi}$ for $D$ in Eq. (8).

We consider a lower bound of measurement errors when we indirectly measure an observable $A_{S}$ of $S$ by measuring a meter observable $M_{D}$ of $D$ under the energy conservation law,

We assume that the meter observable satisfies the Yanase condition:

This means that we can read off the value of the meter observable with zero error under the energy conservation. We define the error of an indirect measurement of $A_{S}$, $\varepsilon(A_{S})$ by

where $\tilde{A_{S}}=I_{O_{\rm I}}\otimes A_{S}\otimes I_{O_{\rm II}}\otimes I_{D}$ and $\tilde{M_{D}}=I_{O_{\rm I}}\otimes I_{S}\otimes I_{O_{\rm II}}\otimes M_{D}$. Under the above assumptions, for orbital states $\ket{\psi}$ and $\ket{\phi}$ which satisfy Eq. (8) and for arbitrary spin states $\ket{\chi}$ and for an arbitrary but fixed state $\ket{\xi}$, the following inequality for $\varepsilon(A_{S})$ holds under the energy conservation:

where $\sigma(H_{\rm I})$ and $\sigma(H_{\rm II})$ are standard deviations of $H_{\rm I}$ and $H_{\rm II}$ on the initial state, respectively, and $\braket{[I_{o_{\rm I}}\otimes A_{S},H_{\rm I}]}$ represents the expectation value of the commutator $[I_{o_{\rm I}}\otimes A_{S},H_{\rm I}]$ on $\ket{\psi}\ket{\chi}$. From Eq. (12), we can find that the lower bound of $\varepsilon(A_{S})$ depends on the non-commutativity of $H_{\rm I}$ and $I_{o_{\rm I}}\otimes A_{S}$, and variances of Hamiltonians $H_{\rm I}$ and $H_{\rm II}$ on the initial state. Therefore we can decrease the bound of the measurement error by increasing the energy variance on the initial state. The proof of Eq. (12) is given in appendix A.

Let us consider whether we are able to implement a SWAP gate using nuclear spins of heavy atoms at rest and photons as an application of Eq. (12). We take this example because it is important from the point of view of engineering to transfer information from a nuclear spin to a photon when we want to transmit the results of quantum computation which are encoded in states of nuclear spins by quantum communication using light. We consider $S$ as a nuclear spin degree and $D$ as a photon spin degree. Let us suppose that we apply a magnetic field in the $x$ direction to the nuclear spin, and the photon moves along with the $x$-axis. There are four degrees of freedom. The first is the orbital degree of freedom of the nuclear spin. The second is the spin degree of freedom of the nuclear spin. The third is the photon’s orbital degree of freedom in the $x$ direction. The forth is the spin of the photon. We would like to swap the state of $S$ and the state of $D$. We consider the following Hamiltonian,

where $b$ is a constant that represents the strength of the magnetic field, and $g$ is the coupling constant between the nuclear spin and the photon. In special relativity, the energy of the photon with momentum $\vec{p}$ is given by $c|\vec{p}|$, and we employed natural units, $c=1$. $w(x)$ is a function of the position operator $x$ of the photon, and it takes non-zero values only for regions where the interaction between the nuclear spin and the photon exists. This type of Hamiltonian is known as Coleman-Hepp model [54]. Here, we consider the dipole- dipole interaction between S and D. This interaction has three-dimensional rotational symmetry, and the strength of the interaction $w(x)$ is able to be arbitrarily set. We consider that Eq. (15) is an effective Hamiltonian model which is applicable to experiments. Fig. 1 is the conceptual figure of the setting. The gray area in the figure represents the region where $w(x)\neq 0$. At first, the nuclear spin and the photon are spatially separated, and there is no interaction. When the photon approaches the nuclear spin, they begin to interact with each other. After the scattering, the photon propagates infinity, and the interaction vanishes. When $g$ is very large, $H_{\rm I}$ and $H_{\rm II}$ are negligible compared to $H_{int}$ in the region where the interaction exists. In the region where $w(x)$ is able to be approximated by a constant function, we are able to make $e^{-iH_{int}t}$ be same as SWAP gate which acts on the spin state spaces of $S$ and $D$ by adjusting $gw(x)$.

We regard the implementation of SWAP gate as the measurement of a nuclear spin $\sigma_{z}$ where a meter observable is a photon spin $\sigma_{z}$ and use Eq. (12), then we find that

holds. Note that because $\sigma_{z,D}$ and $p_{x}$ are operators acting on different Hilbert spaces, the Yanase condition is satisfied. On the other hand, when we succeed in implementing SWAP gate $U_{SWAP}$ perfectly,

must be satisfied. Therefore, we are not able to implement SWAP gate perfectly when the fluctuation of momentum of the photon in the initial state is finite.

We are able to quantify it by deriving an inequality between the gate fidelity of SWAP gate $F_{SWAP}$ and the energy variance. Before showing the inequality of $F_{SWAP}$, we give the definition of the gate fidelity and the worst error probability of the physical realization. We consider that we would like to implement a unitary gate $U_{ideal}$ which acts on two systems $S$ and $D$. We represent the initial state of the composite system $S$+$D$ by $\rho$. We also consider an external system $E$ and suppose that its initial state is $\ket{\xi}$. Considering the purification, we are able to assume that the initial state of the external system is a pure state. We define a unitary time evolution operator which acts on the composite system of $S$+$D$+$E$ by $U$. The state of $S$+$D$ after $U$ is applied, which we represent by $\mathcal{E}(\rho)$, is

On the other hand, if we succeed in implementing $U_{ideal}$ perfectly, the final state of $S$+$D$ is $U_{ideal}\rho U_{ideal}^{\dagger}$. Therefore, we define the worst error probability of the physical realization of $U_{ideal}$ by the completely bounded distance (CB distance) [55, 56] as follows:

where $E^{\prime}$ is another external system and $D(\rho_{1},\rho_{2})$ is the trace distance given by

The gate trace distance between $U_{ideal}$ and the physical implementation $\mathcal{E}$, $D(\mathcal{E},U_{ideal})$ is defined by

From the definition of the CB distance,

holds. By minimizing over all physical implementation $(U,\ket{\xi})$,

can be found. Because of the joint convexity of the trace distance, we are able to assume that $\rho_{S+D}$ is a pure state. The gate fidelity between $U_{ideal}$ and $\mathcal{E}$, which is denoted by $F(\mathcal{E},U_{ideal})$, is give by

where $\ket{\psi}$ is a spin wavefunction of $S+D$ and

The following relation between the trace distance and fielity holds:

Therefore,

is verified. If $D_{CB}(\mathcal{E},U_{ideal})>0$, we are not able to implement $U_{ideal}$ perfectly.

When the Hamiltonian is given by Eq. (15) , for position states which satisfy Eq.(8),

holds. The derivation of this inequality is given in Appendix B.

Furthermore, we are able to extend the bound for the Hadamard gate under an additive conservation law in Ref. [20] to that under the energy conservation law. For example, let us consider we would like to implement the Hadamard gate using the composite system $S$+$D$ where the Hamiltonian of the composite system is

where $s_{x,S}$ and $L_{x,D}$ are the $x$ component of the angular momentum of each system. When the interaction term satisfies Eq. (8) , from Eq. (12) and similar calculations as in Ref. [20], we are able to show the following bound:

where $F_{H}$ is the gate fidelity of the Hadamard gate.

In this section, we show the relation Eq. (12) between the measurement error in scattering processes and the energy conservation. We also prove that SWAP gate is not able to be implemented perfectly as an application of Eq. (12). Furthermore, the bound of the gate fidelity of the Hadamard gate proved in Ref. [20] under an additive conservation law is able to be expanded under the energy conservation in scattering processes.

In this section, we show necessary conditions that the Hamiltonian of a control system must satisfy for implementing controlled unitary gates with zero error in scattering processes under energy conservation. We also give an upper bound of the gate fidelity of two-qubit controlled gates.

Let us represent a control system and a target system as $C$ and $T$, respectively. We consider two orthogonal states of $C$, $\ket{\phi_{0}}_{C}$ and $\ket{\phi_{1}}_{C}$. Note that $\{\ket{\phi_{0}}_{C},\ket{\phi_{1}}_{C}\}$ is not necessarily a complete set. We deal with a controlled gate $U_{CU}$ such that when the state of $C$ is $\ket{\phi_{0}}_{C}$, there is no effect on $T$, and when it is $\ket{\phi_{1}}_{C}$, a unitary operator $V_{T}$ is acting in $T$. The action of $U_{CU}$ is written as

for an arbitrary state $\ket{\psi}_{T}$ of T. Note that final states of $C$, $\ket{\phi_{0}^{\prime}}_{C}$ and $\ket{\phi_{1}^{\prime}}_{C}$ are independent of the initial state of $T$, $\ket{\psi}_{T}$. We consider a non-trivial gate such that $V_{T}\neq I_{T}$.

Let $H_{C}$, $H_{T}$, and $H_{int}$ denote the Hamiltonian of $C$, that of $T$, and an interaction Hamiltonian between $C$ and $T$. The total Hamiltonian of the composite system $C$ and $T$, represented as $H_{C+T}$, is given by

Similar to the previous section, we examine a scattering process in which C and T are initially spatially separated, and the interaction occurs as C approaches T. After some time, C and T become spatially separated once again. We define $\tau$ as the time interval during which the interaction is non-zero. The time evolution operator describing the scattering process $U_{CT}$ is written as

We assume that

for an arbitrary superposition state of $C$, $\ket{\phi}_{C}=c_{0}\ket{\phi_{0}}_{C}+c_{1}\ket{\phi_{1}}_{C}$ where $|c_{0}|^{2}+|c_{1}|^{2}=1$, and an arbitrary state of $T$, $\ket{\psi}_{T}$.

When implementing the controlled unitary gate, $U_{CU}$ using this scattering process, we adjust the interaction $H_{int}$ and the interaction time $\tau$ to satisfy the following relations:

for arbitrary initial state $\ket{\psi}_{T}$ of $T$. However when the energy conservation law

holds, $H_{C}$ must satisfy the following initial condition,

to implement the gate $U_{CU}$ with zero-error.

If $H_{int}$ satisfies the conditions below similar to conditions Eq. (8):

for an arbitrary state $\ket{\psi}_{T}$, the additional condition,

is requested to implement $U_{CU}$ perfectly under energy conservation. Note that if Eq. (38) holds, Eq. (34) is also satisfied. The converse is not true. Thus Eq. (38) is the stronger condition than Eq. (34). When we assume Eq. (38) , we are able to show a more stringent condition Eq.(39) in addition to Eq. (37) . We give the proofs of Eq. (37) and Eq. (39) in appendix C.

Furthermore, if $C$ and $T$ are one-qubit systems, $U_{CU}$ becomes a two-qubit controlled gate, and $V_{T}$ becomes a one-qubit gate. The gate $V_{T}$ can be represented as

where $\vec{u}$ is a three-dimensional real vector and $\vec{\sigma}$ is the vector defined by $\vec{\sigma}\coloneqq(\sigma_{x},\sigma_{y},\sigma_{z})$, where $\sigma_{x}$, $\sigma_{y}$ and $\sigma_{z}$ are the Pauli matrices which act on the Hilbert space of $T$. The domain of $\theta$ and $\phi$ are $0\leq\phi<2\pi$ and $0\leq\theta\leq\pi$, respectively.

When implementing a two-qubit controlled unitary gate $U_{CU}$ using a scattering process under energy conservation, we have the following inequality regarding the CB distance between $U_{CU}$ and the physical implementation $\mathcal{E}_{\alpha}$:

where $\sigma(\vec{l})\coloneqq\ket{\phi_{0}}\bra{\phi_{0}}-\ket{\phi_{1}}\bra{\phi_{1}}$, $\gamma\coloneqq\max\{\|H_{C}\|,\|H_{T}\|\}$ and $H_{A}$ is the free Hamiltonian of the ancillary system $A$. If $\braket{\phi_{0}}{H_{C}}{\phi_{1}}=0$ holds, the Hamiltonian $H_{C}$ commutes with $\sigma(\vec{l})$ and the upper bound of gate fidelity is 1, that is to say, we can implement $U_{CU}$ perfectly. We also find that the bound dependence of the angle $\theta$ is $\sin^{2}\frac{\theta}{2}$. The derivation of Eq. (41) is provided in appendix D.

In this section, we show that $H_{C}$ must satisfy Eq. (37) and Eq. (39) to implement a non-trivial controlled gate whose action is given by Eq.(31) with zero-error using scattering process under energy conservation law. We also obtain the upper bound of gate fidelity Eq. (41) for a two-qubit controlled gate.

We showed a lower bound of the measurement error Eq. (12) for scattering-type measurements which satisfy Eq. (8) under the energy conservation in section 2. In section 2 we also presented that the SWAP gate is not able to be implemented perfectly as an application of Eq. (12). Futhermore, the bound of the gate fidelity of the Hadamard gate proved in Ref. [20] under an additive conservation law is able to be expanded under the energy conservation in scattering processes. In section 3, we gived the necessary condition Eq. (37) which must be fulfilled to implement a controlled unitary gate described by Eq. (31) without error in a scattering process under energy conservation. We also proved that when the interaction term satisfies Eq. (38), the additional necessary condition for $H_{C}$, Eq. (39) must also be fulfilled to implement a controlled unitary gate perfectly. Moreover, when $C$ and $T$ are one-qubit systems, the quantitative relation Eq. (41) between the gate fidelity and the non-diagonal entry of $H_{C}$ is shown. This result extends Ozawa’s result which evaluates the upper bound of the gate fidelity of CNOT gates [21].