Showing papers in "Physical Review A in 2016"
IBM1
TL;DR: In this article, the authors present improvements in both theoretical understanding and experimental implementation of the cross resonance (CR) gate that have led to shorter two-qubit gate times and interleaved randomized benchmarking fidelities exceeding 99%.
Abstract: We present improvements in both theoretical understanding and experimental implementation of the cross resonance (CR) gate that have led to shorter two-qubit gate times and interleaved randomized benchmarking fidelities exceeding 99%. The CR gate is an all-microwave two-qubit gate that does not require tunability and is therefore well suited to quantum computing architectures based on two-dimensional superconducting qubits. The performance of the gate has previously been hindered by long gate times and fidelities averaging 94--96%. We have developed a calibration procedure that accurately measures the full CR Hamiltonian. The resulting measurements agree with theoretical analysis of the gate and also elucidate the error terms that have previously limited gate fidelity. The increase in fidelity that we have achieved was accomplished by introducing a second microwave drive tone on the target qubit to cancel unwanted components of the CR Hamiltonian.
447 citations
TL;DR: In this paper, a general protocol to measure out-of-time-order correlation functions is proposed for diagnosing the scrambling of quantum information in interacting quantum systems and has recently received particular attention in the study of chaos and black holes within holographic duality.
Abstract: We provide a general protocol to measure out-of-time-order correlation functions. These correlation functions are of broad theoretical interest for diagnosing the scrambling of quantum information in interacting quantum systems and have recently received particular attention in the study of chaos and black holes within holographic duality. Measuring them requires an echo-type sequence in which the sign of a many-body Hamiltonian is reversed. We illustrate our protocol by detailing an implementation employing cold atoms and cavity quantum electrodynamics to probe spin models with nonlocal interactions. To verify the feasibility of the scheme with current technology, we analyze the effects of dissipation in a chaotic kicked-top model. Finally, we propose a number of other experimental platforms where similar out-of-time-order correlation functions can be measured.
436 citations
TL;DR: This work proposes a method for introducing independent random single-qubit gates into the logical circuit in such a way that the effective logical circuit remains unchanged and proves that this randomization tailors the noise into stochastic Pauli errors, which can dramatically reduce error rates while introducing little or no experimental overhead.
Abstract: Quantum computers are poised to radically outperform their classical counterparts by manipulating coherent quantum systems. A realistic quantum computer will experience errors due to the environment and imperfect control. When these errors are even partially coherent, they present a major obstacle to performing robust computations. Here, we propose a method for introducing independent random single-qubit gates into the logical circuit in such a way that the effective logical circuit remains unchanged. We prove that this randomization tailors the noise into stochastic Pauli errors, which can dramatically reduce error rates while introducing little or no experimental overhead. Moreover, we prove that our technique is robust to the inevitable variation in errors over the randomizing gates and numerically illustrate the dramatic reductions in worst-case error that are achievable. Given such tailored noise, gates with significantly lower fidelity---comparable to fidelities realized in current experiments---are sufficient to achieve fault-tolerant quantum computation. Furthermore, the worst-case error rate of the tailored noise can be directly and efficiently measured through randomized benchmarking protocols, enabling a rigorous certification of the performance of a quantum computer.
331 citations
TL;DR: In this paper, a general theory of sensors based on the detection of splittings of resonant frequencies or energy levels operating at so-called exceptional points is presented, where the complex-square-root topology near such non-Hermitian degeneracies has a great potential for enhanced sensitivity.
Abstract: A general theory of sensors based on the detection of splittings of resonant frequencies or energy levels operating at so-called exceptional points is presented. Exploiting the complex-square-root topology near such non-Hermitian degeneracies has a great potential for enhanced sensitivity. Passive and active systems are discussed. The theory is specified for whispering-gallery microcavity sensors for particle detection. As example, a microdisk with two holes is studied numerically. The theory and numerical simulations demonstrate a sevenfold enhancement of the sensitivity.
327 citations
TL;DR: In this paper, a general framework for studying quantum coherence as a resource is presented, where coherence is classified as "speakable" or "unspeakable", depending on whether or not the identity of the subspaces that appear in the coherent superposition is significant.
Abstract: A general framework for studying quantum coherence as a resource is presented, where coherence is classified as ``speakable'' or ``unspeakable'' based on whether or not the identity of the subspaces that appear in the coherent superposition is significant.
311 citations
TL;DR: In this article, the authors exploit the different modes of a silicon ring resonator as an extra dimension for photons to generate topologically robust optical isolators and driven-dissipative analog of the 4D quantum Hall effect.
Abstract: Recent technological advances in integrated photonics have spurred on the study of topological phenomena in engineered bosonic systems. Indeed, the controllability of silicon ring-resonator arrays has opened up new perspectives for building lattices for photons with topologically nontrivial bands and integrating them into photonic devices for practical applications. Here, we push these developments even further by exploiting the different modes of a silicon ring resonator as an extra dimension for photons. Tunneling along this synthetic dimension is implemented via an external time-dependent modulation that allows for the generation of engineered gauge fields. We show how this approach can be used to generate a variety of exciting topological phenomena in integrated photonics, ranging from a topologically-robust optical isolator in a spatially one-dimensional (1D) ring-resonator chain to a driven-dissipative analog of the 4D quantum Hall effect in a spatially 3D resonator lattice. Our proposal paves the way towards the use of topological effects in the design of novel photonic lattices supporting many frequency channels and displaying higher connectivities.
308 citations
TL;DR: In this paper, the authors demonstrate measurement times exceeding 10 − 5 s and zeptonewton force sensitivity with laser-cooled silica nanospheres trapped in an optical lattice, which enables a variety of applications including electric-field sensing, inertial sensing, and gravimetry.
Abstract: Optically trapped nanospheres in high vacuum experience little friction and hence are promising for ultrasensitive force detection. Here we demonstrate measurement times exceeding ${10}^{5}$ s and zeptonewton force sensitivity with laser-cooled silica nanospheres trapped in an optical lattice. The sensitivity achieved exceeds that of conventional room-temperature solid-state force sensors by over an order of magnitude, and enables a variety of applications including electric-field sensing, inertial sensing, and gravimetry. The particle is confined at the antinodes of the optical standing wave, and by studying the motion of a particle which has been moved to an adjacent trapping site, the known spacing of the antinodes can be used to calibrate the displacement spectrum of the particle. Finally, we study the dependence of the trap stability and lifetime on the laser intensity and gas pressure, and examine the heating rate of the particle in vacuum without feedback cooling.
293 citations
TL;DR: It is shown that trace distance measure of coherence is a strong monotone for all qubit and, so called, $X$ states and an expression for the trace distance coherence for all pure states and a semi definite program for arbitrary states is provided.
Abstract: We show that trace distance measure of coherence is a strong monotone for all qubit and, so called, $X$ states. An expression for the trace distance coherence for all pure states and a semidefinite program for arbitrary states is provided. We also explore the relation between ${l}_{1}$-norm and relative entropy based measures of coherence, and give a sharp inequality connecting the two. In addition, it is shown that both ${l}_{p}$-norm- and Schatten-$p$-norm--based measures violate the (strong) monotonicity for all $p\ensuremath{\in}(1,\ensuremath{\infty})$.
265 citations
TL;DR: This work presents a four-intensity protocol for the decoy-state MDI-QKD that hugely raises the key rate, especially in the case in which the total data size is not large.
Abstract: The relatively low key rate seems to be the major barrier to its practical use for the decoy-state measurement-device-independent quantum key distribution (MDI-QKD). We present a four-intensity protocol for the decoy-state MDI-QKD that hugely raises the key rate, especially in the case in which the total data size is not large. Also, calculations show that our method makes it possible for secure private communication with fresh keys generated from MDI-QKD with a delay time of only a few seconds.
250 citations
TL;DR: In this paper, the robustness of asymmetry of quantum states has been studied in the context of phase discrimination and quantum metrology, and the robustity of quantum coherence has been analyzed.
Abstract: Quantum states may exhibit asymmetry with respect to the action of a given group. Such an asymmetry of states can be considered as a resource in applications such as quantum metrology, and it is a concept that encompasses quantum coherence as a special case. We introduce explicitly and study the robustness of asymmetry, a quantifier of asymmetry of states that we prove to have many attractive properties, including efficient numerical computability via semidefinite programming, and an operational interpretation in a channel discrimination context. We also introduce the notion of asymmetry witnesses, whose measurement in a laboratory detects the presence of asymmetry. We prove that properly constrained asymmetry witnesses provide lower bounds to the robustness of asymmetry, which is shown to be a directly measurable quantity itself. We then focus our attention on coherence witnesses and the robustness of coherence, for which we prove a number of additional results; these include an analysis of its specific relevance in phase discrimination and quantum metrology, an analytical calculation of its value for a relevant class of quantum states, and tight bounds that relate it to another previously defined coherence monotone.
247 citations
TL;DR: In this paper, the authors consider the problem of simultaneous estimation of multiple parameters in quantum metrological models and show that for every estimated parameter, the variance obtained in the multiparameter scheme is equal to that of an optimal scheme for that parameter alone, assuming all other parameters are perfectly known.
Abstract: Simultaneous estimation of multiple parameters in quantum metrological models is complicated by factors relating to the (i) existence of a single probe state allowing for optimal sensitivity for all parameters of interest, (ii) existence of a single measurement optimally extracting information from the probe state on all the parameters, and (iii) statistical independence of the estimated parameters. We consider the situation when these concerns present no obstacle, and for every estimated parameter the variance obtained in the multiparameter scheme is equal to that of an optimal scheme for that parameter alone, assuming all other parameters are perfectly known. We call such models compatible. In establishing a rigorous theoretical framework for investigating compatibility, we clarify some ambiguities and inconsistencies present in the literature and discuss several examples to highlight interesting features of unitary and nonunitary parameter estimation, as well as deriving new bounds for physical problems of interest, such as the simultaneous estimation of phase and local dephasing.
TL;DR: In this paper, Zhao et al. showed that dipole-driven collapse induced by soft excitations is compensated by the repulsive Lee-Huang-Yang contribution resulting from quantum fluctuations of hard excitations, in a similar mechanism as that recently proposed for Bose-Bose mixtures.
Abstract: Collapse in dipolar Bose-Einstein condensates may be arrested by quantum fluctuations. Due to the anisotropy of the dipole-dipole interactions, the dipole-driven collapse induced by soft excitations is compensated by the repulsive Lee-Huang-Yang contribution resulting from quantum fluctuations of hard excitations, in a similar mechanism as that recently proposed for Bose-Bose mixtures. The arrested collapse results in self-bound filamentlike droplets, providing an explanation for the intriguing results of recent dysprosium experiments. Arrested instability and droplet formation are general features directly linked to the nature of the dipole-dipole interactions, and should hence play an important role in all future experiments with strongly dipolar gases.
TL;DR: An algorithm for prediction on a quantum computer which is based on a linear regression model with least-squares optimization, which is adapted to process nonsparse data matrices that can be represented by low-rank approximations, and significantly improve the dependency on its condition number.
Abstract: We give an algorithm for prediction on a quantum computer which is based on a linear regression model with least-squares optimization. In contrast to related previous contributions suffering from the problem of reading out the optimal parameters of the fit, our scheme focuses on the machine-learning task of guessing the output corresponding to a new input given examples of data points. Furthermore, we adapt the algorithm to process nonsparse data matrices that can be represented by low-rank approximations, and significantly improve the dependency on its condition number. The prediction result can be accessed through a single-qubit measurement or used for further quantum information processing routines. The algorithm's runtime is logarithmic in the dimension of the input space provided the data is given as quantum information as an input to the routine.
TL;DR: In this paper, a number of coherence measures are introduced based on relative R\'enyi entropies, and the incoherent Schmidt rank can be increased arbitrarily large by certain noncoherence generating operations.
Abstract: A resource theory of quantum coherence attempts to characterize the quantum coherence that exists in a given quantum system. Many different approaches to a resource theory of coherence have recently been proposed, with their differences lying primarily in the identification of ``free'' or ``incoherent'' operations. In this article, we compare a number of these operational classes. In particular, the recently introduced class of dephasing-covariant operations is analyzed, and we characterize the Kraus operators of such maps. A number of coherence measures are introduced based on relative R\'enyi entropies, and we study incoherent state transformations under different operational classes. In particular, we show that the incoherent Schmidt rank can be increased arbitrarily large by certain noncoherence generating operations. The distinction between asymmetry-based versus basis-dependent notions of coherence theory is clarified, and we further develop the resource theory of $N$ asymmetry, where $N$ is the group of all diagonal incoherent unitaries.
TL;DR: A systematic study assessing the impact of the effective temperatures in the learning of a special class of a restricted Boltzmann machine embedded on quantum hardware, which can serve as a building block for deep-learning architectures.
Abstract: An increase in the efficiency of sampling from Boltzmann distributions would have a significant impact on deep learning and other machine-learning applications. Recently, quantum annealers have been proposed as a potential candidate to speed up this task, but several limitations still bar these state-of-the-art technologies from being used effectively. One of the main limitations is that, while the device may indeed sample from a Boltzmann-like distribution, quantum dynamical arguments suggest it will do so with an instance-dependent effective temperature, different from its physical temperature. Unless this unknown temperature can be unveiled, it might not be possible to effectively use a quantum annealer for Boltzmann sampling. In this work, we propose a strategy to overcome this challenge with a simple effective-temperature estimation algorithm. We provide a systematic study assessing the impact of the effective temperatures in the learning of a special class of a restricted Boltzmann machine embedded on quantum hardware, which can serve as a building block for deep-learning architectures. We also provide a comparison to $k$-step contrastive divergence (CD-$k$) with $k$ up to 100. Although assuming a suitable fixed effective temperature also allows us to outperform one-step contrastive divergence (CD-1), only when using an instance-dependent effective temperature do we find a performance close to that of CD-100 for the case studied here.
TL;DR: In this article, the Deutsch-Jozsa algorithm is applied to a probabilistic version of the decision problem, and the quantum probability of being correct for both classical and quantum procedures is compared.
Abstract: That superpositions of states can be useful for performing tasks in quantum systems has been known since the early days of quantum information, but only recently has a quantitative theory of quantum coherence been proposed. Here we apply that theory to an analysis of the Deutsch-Jozsa algorithm, which depends on quantum coherence for its operation. The Deutsch-Jozsa algorithm solves a decision problem, and we focus on a probabilistic version of that problem, comparing probability of being correct for both classical and quantum procedures. In addition, we study a related decision problem in which the quantum procedure has one-sided error while the classical procedure has two-sided error. The role of coherence on the quantum success probabilities in both of these problems is examined.
TL;DR: In this article, it is shown that speed is related to the notion of asymmetry, a result that provides direct connection with the problem of quantifying the coherence of quantum states.
Abstract: Speed is a notion that formally sets bounds on the time that a quantum system takes to evolve. Here it is shown that speed is related to the notion of asymmetry, a result that provides direct connection with the problem of quantifying the coherence of quantum states.
TL;DR: In this article, a self-bound dilute quantum gaseous dipolar Bose-Einstein condensate was proposed to produce a self bound dilute QGAS.
Abstract: A liquid droplet is a self-bound phase of matter that holds itself together in the absence of a container. Without a container a gas will normally expand to fill space. A method is proposed to produce a self-bound dilute quantum gaseous dipolar Bose-Einstein condensate.
TL;DR: In this paper, the ground state of dipolar Bose-Einstein condensates (BEC) is studied in the regimes of three different phases, where a characteristic growth of the peak density of the BEC is predicted when the dipolar quantum droplets are formed.
Abstract: The ground state of dipolar Bose-Einstein condensates (BEC) is studied in the regimes of three different phases, where a characteristic growth of the peak density of the BEC is predicted when the dipolar quantum droplets are formed.
TL;DR: In this paper, the authors proposed a protocol to measure the time dependence of out-of-time-ordered (OTO) correlation functions using an ancilla that controls the direction of time.
Abstract: There has been recent progress in understanding chaotic features in many-body quantum systems. Motivated by the scrambling of information in black holes, it has been suggested that the time dependence of out-of-time-ordered (OTO) correlation functions such as $\ensuremath{\langle}{O}_{2}(t){O}_{1}(0){O}_{2}(t){O}_{1}(0)\ensuremath{\rangle}$ is a faithful measure of quantum chaos. Experimentally, these correlators are challenging to access since they apparently require access to both forward and backward time evolution with the system Hamiltonian. Here we propose a protocol to measure such OTO correlators using an ancilla that controls the direction of time. Specifically, by coupling the state of the ancilla to the system Hamiltonian of interest, we can emulate the forward and backward time propagation, where the ancilla plays the role of a quantum clock. Within this scheme, the continuous evolution of the entire system (the system of interest and the ancilla) is governed by a time-independent Hamiltonian. We discuss the implementation of our protocol with current circuit-QED technology for a class of interacting Hamiltonians. Our protocol is immune to errors that could occur when the direction of time evolution is externally controlled by a classical switch.
TL;DR: In this article, the ground state properties of a trapped dipolar condensate under the influence of quantum fluctuations were investigated and it was shown that this system can undergo a phase transition from a low-density Condensate state to a high-density droplet state, which is stabilized by quantum fluctuations.
Abstract: We consider the ground state properties of a trapped dipolar condensate under the influence of quantum fluctuations. We show that this system can undergo a phase transition from a low density condensate state to a high density droplet state, which is stabilized by quantum fluctuations. The energetically favored state depends on the geometry of the confining potential, the number of atoms, and the two-body interactions. We develop a simple variational ansatz and validate it against full numerical solutions. We produce a phase diagram for the system and present results relevant to current experiments with dysprosium and erbium condensates.
TL;DR: In this article, it was shown that the nonlocal nonlinear Schrodinger equation recently proposed by Ablowitz and Musslimani [Phys. Rev. Lett. 110, 064105 (2013)] is gauge equivalent to the system of coupled Landau-Lifshitz equations.
Abstract: It is shown that the nonlocal nonlinear Schr\"odinger equation recently proposed by Ablowitz and Musslimani [Phys. Rev. Lett. 110, 064105 (2013)] is gauge equivalent to the unconventional system of coupled Landau-Lifshitz equations. The first integrals of motion and one-soliton solution of an obtained model are given. The physical and geometrical aspects of model and their effect on expected metamagnetic structures are studied.
TL;DR: This paper utilises the IBM chip to realise protocols in Quantum Error Correction, Quantum Arithmetic, Quantum graph theory and Fault-tolerant quantum computation, by accessing the device remotely through the cloud.
Abstract: Quantum computing technology has reached a second renaissance in the past five years. Increased interest from both the private and public sector combined with extraordinary theoretical and experimental progress has solidified this technology as a major advancement in the 21st century. As anticipated my many, some of the first realizations of quantum computing technology has occured over the cloud, with users logging onto dedicated hardware over the classical internet. Recently, IBM has released the Quantum Experience, which allows users to access a five-qubit quantum processor. In this paper we take advantage of this online availability of actual quantum hardware and present four quantum information experiments. We utilize the IBM chip to realize protocols in quantum error correction, quantum arithmetic, quantum graph theory, and fault-tolerant quantum computation by accessing the device remotely through the cloud. While the results are subject to significant noise, the correct results are returned from the chip. This demonstrates the power of experimental groups opening up their technology to a wider audience and will hopefully allow for the next stage of development in quantum information technology.
IBM1
TL;DR: In this article, a method for distinguishing between unitary and non-unitary errors in quantum gates by interleaving repetitions of a target gate within a randomized benchmarking sequence is presented.
Abstract: With improved gate calibrations reducing unitary errors, we achieve a benchmarked single-qubit gate fidelity of $0.9995\ifmmode\pm\else\textpm\fi{}0.0002$ with superconducting qubits in a circuit quantum electrodynamics system. We present a method for distinguishing between unitary and nonunitary errors in quantum gates by interleaving repetitions of a target gate within a randomized benchmarking sequence. The benchmarking fidelity decays quadratically with the number of interleaved gates for unitary errors but linearly for nonunitary errors, allowing us to separate systematic coherent errors from decoherent effects. With this protocol, we show that the fidelity of the gates is not limited by unitary errors.
TL;DR: In this paper, an alternative framework for quantifying coherence is proposed, based on the additivity of coherence for subspace-independent states, which is described by an operationindependent equality rather than operation-dependent inequalities and therefore applicable to various physical contexts.
Abstract: We propose an alternative framework for quantifying coherence. The framework is based on a natural property of coherence, the additivity of coherence for subspace-independent states, which is described by an operation-independent equality rather than operation-dependent inequalities and therefore applicable to various physical contexts. Our framework is compatible with all the known results on coherence measures but much more flexible and convenient for applications, and by using it many open questions can be resolved.
TL;DR: In this paper, the authors studied coherence measures induced by quantum divergences of the Tsallis type and derived the trade-off relations between coherence and mixedness.
Abstract: The concept of coherence is one of cornerstones in physics. The development of quantum information science has lead to renewed interest in properly approaching the coherence at the quantum level. Various measures could be proposed to quantify coherence of a quantum state with respect to the prescribed orthonormal basis. To be a proper measure of coherence, each candidate should enjoy certain properties. It seems that the monotonicity property plays a crucial role here. Indeed, there is known an intuitive measure of coherence that does not share this condition. We study coherence measures induced by quantum divergences of the Tsallis type. Basic properties of the considered coherence quantifiers are derived. Trade-off relations between coherence and mixedness are examined. The property of monotonicity under incoherent selective measurements has to be reformulated. The proposed formulation can naturally be treated as a parametric extension of its standard form. Finally, two coherence measures quadratic in moduli of matrix elements are compared from the monotonicity viewpoint.
TL;DR: In this paper, the properties of atom-photon bound states in waveguide QED systems consisting of single or multiple atoms coupled strongly to a finite-bandwidth photonic channel are discussed.
Abstract: We discuss the properties of atom-photon bound states in waveguide QED systems consisting of single or multiple atoms coupled strongly to a finite-bandwidth photonic channel. Such bound states are formed by an atom and a localized photonic excitation and represent the continuum analog of the familiar dressed states in single-mode cavity QED. Here we present a detailed analysis of the linear and nonlinear spectral features associated with single- and multiphoton dressed states and show how the formation of bound states affects the waveguide-mediated dipole-dipole interactions between separated atoms. Our results provide both a qualitative and quantitative description of the essential strong-coupling processes in waveguide QED systems, which are currently being developed in the optical and microwave regimes.
TL;DR: In this article, the authors investigate the dynamics following a global parameter quench for two one-dimensional models with variable-range power-law interactions: a long-range transverse Ising model, which has recently been realized in chains of trapped ions, and a longrange lattice model for spinless fermions with long-term tunneling.
Abstract: We investigate the dynamics following a global parameter quench for two one-dimensional models with variable-range power-law interactions: a long-range transverse Ising model, which has recently been realized in chains of trapped ions, and a long-range lattice model for spinless fermions with long-range tunneling. For the transverse Ising model, the spreading of correlations and growth of entanglement are computed using numerical matrix product state techniques, and are compared with exact solutions for the fermionic tunneling model. We identify transitions between regimes with and without an apparent linear light cone for correlations, which correspond closely between the two models. For long-range interactions, we find that despite the lack of a light cone, correlations grow slowly as a power law at short times, and that—depending on the structure of the initial state—the growth of entanglement can also be sublinear. These results are understood through analytical calculations, and should be measurable in experiments with trapped ions.
TL;DR: In this paper, the authors derived the microscopic optomagnonic Hamiltonian of a macrospin in the optical cavities and showed that the induced dissipation coefficient can change sign on the Bloch sphere, leading to self-sustained oscillations.
Abstract: Experiments during the past 2 years have shown strong resonant photon-magnon coupling in microwave cavities, while coupling in the optical regime was demonstrated very recently for the first time. Unlike with microwaves, the coupling in optical cavities is parametric, akin to optomechanical systems. This line of research promises to evolve into a new field of optomagnonics, aimed at the coherent manipulation of elementary magnetic excitations in solid-state systems by optical means. In this work we derive the microscopic optomagnonic Hamiltonian. In the linear regime the system reduces to the well-known optomechanical case, with remarkably large coupling. Going beyond that, we study the optically induced nonlinear classical dynamics of a macrospin. In the fast-cavity regime we obtain an effective equation of motion for the spin and show that the light field induces a dissipative term reminiscent of Gilbert damping. The induced dissipation coefficient, however, can change sign on the Bloch sphere, giving rise to self-sustained oscillations. When the full dynamics of the system is considered, the system can enter a chaotic regime by successive period doubling of the oscillations.
TL;DR: Three circuit identities are reported that provide the means for replacing certain configurations of the multiple control Toffoli gates with their simpler relative phase implementations, up to a selectable unitary on certain qubits, and without changing the overall functionality.
Abstract: Various implementations of the Toffoli gate up to a relative phase have been known for years. The advantage over the regular Toffoli gate is its smaller circuit size. However, its use has been often limited to a demonstration of quantum control in designs such as those where the Toffoli gate is being applied last or otherwise for some specific reasons the relative phase does not matter. It was commonly believed that the relative-phase deviations would prevent the relative-phase Toffoli gates from being very helpful in practical large-scale designs. In this paper, we report three circuit identities that provide the means for replacing certain configurations of the multiple control Toffoli gates with their simpler relative-phase implementations, up to a selectable unitary on certain qubits, and without changing the overall functionality. We illustrate the advantage via applying those identities to the optimization of the known circuits implementing multiple control Toffoli gates, and report the reductions in the controlled-not count, $T$count, as well as the number of ancillae used. We suggest that a further study of the relative-phase Toffoli implementations and their use may yield other optimizations.