Parallel multi-qubit operations on a universal trapped ion quantum computer

Abstract

The present disclosure describes various aspects related to implementing efficient multi-qubit operations, and more specifically, relates to techniques for implementing parallel multi-qubit operations on a general ion trap quantum computer. In one aspect, a method for performing quantum operations in an ion trap quantum computer or a trapped ion quantum system includes implementing at least two parallel gates of a quantum circuit, each of the at least two parallel gates being a multi-qubit gate, each of the at least two parallel gates being implemented using a different set of ions from a plurality of ions in an ion trap, and the plurality of ions comprising four or more ions. The method also includes, as part of the quantum operation, performing operations on the at least two parallel gates simultaneously. A trapped ion quantum system and a computer-readable storage medium corresponding to the above method are also disclosed.

Description

[0001] CROSS-REFERENCE TO RELATED APPLICATIONS

[0002] This application claims priority to and the benefit of U.S. Provisional Patent Application No. 62 / 682,677, filed on June 8, 2018, entitled “PARALLEL 2-QUBIT OPERATIONS ON A UNIVERSAL ION TRAP QUANTUM COMPUTER,” and U.S. Patent Application No. 16 / 433,950, filed on June 6, 2019, entitled “PARALLEL MULTI-QUBIT OPERATIONS ON A UNIVERSAL ION TRAP QUANTUM COMPUTER,” the entireties of which are expressly incorporated herein by reference.

[0003] Government licensing rights

[0004] This invention was made with government support under NSF grant PHY0822671, IARPA grant W911NF1610082, ARO grant W911NF1610349, and AFOSR grant FA95501410052. The government has certain rights in this invention. Technical Field

[0005] Aspects of the present disclosure relate generally to achieving efficient multi-qubit operations, and more particularly, to techniques for achieving parallel multi-qubit operations on a universal ion trap quantum computer. Background Art

[0006] The circuit model of a quantum computer consists of a sequence of gate operations between quantum bits (qubits) that come from a common family of discrete operations, and the ability to execute these gates in parallel is critical to the advancement of quantum computing technology. The ability to execute quantum gates simultaneously or in parallel provides significant efficiency gains for many quantum circuits and algorithms. More importantly, quantum gate parallelism is critical for fault-tolerant error correction of qubits affected by idle decoherence. Without gate parallelism, the error threshold required for fault tolerance may be too small to be achieved for practical systems. The implementation of parallel quantum gates is complicated by potential crosstalk, especially between qubits that are fully connected through a common-mode bus, such as in Coulomb-coupled trapped atomic ions or cavity-coupled superconducting transmons.

[0007] It is therefore desirable to achieve techniques that allow for quantum gate parallelism and avoid some of the existing limitations, particularly as it relates to trapped ion technology. Summary of the invention

[0008] A brief summary of one or more aspects is given below to provide a basic understanding of these aspects. This summary is not an extensive overview of all intended aspects, and is neither intended to identify the key or essential elements of all aspects, nor to delineate the scope of any or all aspects. Its purpose is to present some concepts of one or more aspects in a simplified form as a prelude to a more detailed description that is given later.

[0009] The present disclosure provides various aspects related to multi-qubit operations, and more specifically, provides techniques for implementing parallel multi-qubit (at least two qubit) operations on a universal ion trap quantum computer. The present disclosure describes a method for performing quantum operations in a trapped ion quantum system, the method comprising implementing at least two parallel gates of a quantum circuit, each of the at least two parallel gates being a multi-qubit gate, each of the at least two parallel gates being implemented using a different set of ions from a plurality of ions in an ion trap, and the plurality of ions comprising four or more ions. The method also includes performing operations on the at least two parallel gates simultaneously as part of the quantum operation.

[0010] The present disclosure also describes a trapped ion quantum system configured to perform quantum operations, the trapped ion quantum system comprising: an algorithm component configured to implement at least two parallel gates of a quantum circuit, each of the at least two parallel gates being a multi-qubit gate, each of the at least two parallel gates being implemented using a different set of ions of a plurality of ions in an ion trap, and the plurality of ions comprising four or more ions. The trapped ion quantum system also comprises the ion trap, wherein, as part of the quantum operation, operations are performed simultaneously on the at least two parallel gates. The trapped ion quantum system may also be referred to as a universal ion trap quantum computer, a quantum information processing (QIP) system, a quantum computer, or generally as a computer device.

[0011] The present disclosure also describes a computer-readable storage medium storing code having instructions executable by a processor for performing quantum operations in a trapped ion quantum system, the computer-readable storage medium including code for implementing at least two parallel gates of a quantum circuit, each of the at least two parallel gates being a multi-qubit gate, each of the at least two parallel gates being implemented using a different set of ions of a plurality of ions in an ion trap, and the plurality of ions comprising four or more ions. The computer-readable storage medium also includes code for performing operations on the at least two parallel gates simultaneously as part of the quantum operation.

[0012] Additional details regarding the methods, systems, and computer-readable storage media are described herein for various aspects associated with parallel multi-qubit operations using sets of trapped ions. BRIEF DESCRIPTION OF THE DRAWINGS

[0013] The drawings depict only some embodiments and therefore should not be considered limiting in scope.

[0014] Figure 1 A diagram showing a vacuum chamber housing electrodes for trapping atomic ions in a linear crystal in accordance with aspects of the present disclosure.

[0015] Figure 2A and Figure 2B Examples of pulse shape solutions and theoretical phase space trajectories for parallel XX or Ising gates implemented on ions (1, 4) and (2, 5) of a 5-ion trap are shown in accordance with aspects of the present disclosure.

[0016] Figure 3A and Figure 3B Examples of parity curves and fidelity of parallel XX gates on two exemplary ion sets for a 5-ion trap in accordance with aspects of the present disclosure are shown.

[0017] Figure 4 An example of data for simultaneous CNOT gating on ions (1, 4) and (2, 3) of a 5-ion trap is shown in accordance with aspects of the present disclosure.

[0018] Figure 5A An example of Feynman's original quantum full adder is shown in accordance with aspects of the present disclosure.

[0019] Figure 5B An example of an optimized full adder with a multi-qubit gate depth of 4 is shown in accordance with aspects of the present disclosure.

[0020] Figure 6 An example of data showing a full adder using multi-qubit gates simultaneously on ions (1, 2, 4, 5) of a 5-ion trap in accordance with aspects of the present disclosure.

[0021] Figure 7 FIG. 1 shows a schematic diagram of a multi-qubit system using XX(χ) gates, R with two parallel multi-qubit operations marked in dashed boxes according to aspects of the present disclosure. x (θ) gate and R y Application example of an optimized full adder implementation of the (θ) gate.

[0022] Figure 8 is a diagram illustrating an example of a computer device according to aspects of the present disclosure.

[0023] Fig. 9is a flow chart illustrating an example of a method according to aspects of the present disclosure.

[0024] Fig.10 is a block diagram illustrating an example of a QIP system according to aspects of the present disclosure. DETAILED DESCRIPTION

[0025] The detailed description given below in conjunction with the accompanying drawings is intended as a description of various configurations or embodiments, and is not intended to represent the only configuration or embodiment that can practice the concepts described herein. The detailed description includes specific details for a thorough understanding of the purpose of various concepts. However, it is obvious to those skilled in the art that these concepts can be practiced without these specific details. In some instances, well-known components are shown in block diagram form to avoid blurring these concepts.

[0026] As mentioned above, the circuit model of a quantum computer or quantum information processing (QIP) system consists of a sequence of gate operations between qubits that come from a common family of discrete operations, and the ability to execute these gates in parallel is critical to the advancement of quantum computing technology. The ability to execute quantum gates simultaneously or in parallel provides significant efficiency gains for many quantum circuits and algorithms. More importantly, quantum gate parallelism is essential for error-tolerant correction of qubits that are subject to idle decoherence. Without gate parallelism, the error threshold required for fault tolerance is prohibitively small for practical systems. Since quantum error correction is a particularly heavy process in which the overhead can be severe, a large amount of information needs to be encoded to achieve redundancy. Quantum gate parallelism allows fewer errors to be accumulated, so less redundancy needs to be encoded.

[0027] However, implementing parallel quantum gates is not a simple task. It is often complicated by potential crosstalk, especially between qubits that are fully connected through a common mode bus, such as in Coulomb-coupled trapped atomic ions or cavity-coupled superconducting transporters. In this regard, the present disclosure describes two qubit entanglement gates in parallel in an array of fully connected trapped ion qubits. The application of this capability can be realized in various quantum circuits. An example is the implementation in a quantum full adder using a depth-4 quantum circuit with parallel multi-qubit operations. These results utilize the performance (power) of a highly connected qubit system through classical control techniques, and provide improvements in accelerating quantum circuits and realizing fault tolerance of trapped ion quantum computers. As used herein, the term "multi-qubit" can refer to operations or gates using more than one qubit (such as 2-qubits, 3-qubits, 4-qubits, etc.), including n-qubit operations or gates, where n is an integer.

[0028] Parallel operations save considerable time compared to performing serial operations, improving computational efficiency. Key quantum computing circuits, such as error correction, quantum Fourier transforms, large entangled states, and adders, can benefit from parallelism. Many full quantum algorithms benefit similarly, including Shor's integer factorization, solving the discrete logarithm problem on elliptic curve groups, simulating Hamiltonian dynamics using the Suzuki-Trotter formula, and quantum chemistry algorithms. These improvements are universal and substantial. Both full adders and Toffoli gates can use parallelism to obtain exponential improvements in total execution time. Other quantum algorithms see speedups of O(n) on gate complexities of low-degree polynomials in n.

[0029] The present disclosure describes various techniques for quantum gate parallelism using trapped ions. Trapped atoms and superconducting circuits can be used to realize trapped ion quantum computers, also known as quantum information processing systems. Atom-based qubits can be used as different types of devices, including but not limited to quantum memories, quantum gates in quantum computers and simulators, and nodes for quantum communication networks. Qubits based on trapped atomic ions can have very good coherence, can be prepared and measured with efficiency close to 100%, and can be easily entangled with each other by modulating their Coulomb interactions with suitable external control fields (such as light fields or microwave fields). As used in the present disclosure, the terms "atomic ions", "atoms" and "ions" can be used interchangeably to describe particles that will be confined or actually confined in a trap to form a crystal or similar arrangement or configuration. Also as used in the present disclosure, the terms "gate" and "quantum gate" can be used interchangeably.

[0030] The typical ion trap geometry or structure used for quantum information processing is the linear radio frequency (RF) Paul trap (also called RF trap, surface trap, Paul trap, or simply ion trap), in which nearby electrodes maintain static and dynamic potentials, resulting in effective inhomogeneous harmonic confinement of ions. An RF Paul trap is a trap that uses an electric field to capture or confine charged particles to a specific area, position, or location. When atomic ions are laser-cooled to very low temperatures in such a trap, the external confinement forces are balanced by Coulomb repulsion, and the atomic ions form a stable crystal of qubits (e.g., a structured arrangement of qubits). Due to sufficient trap anisotropy, ions can form linear crystals along weak confinement directions, and this is an arrangement typically used for quantum information and metrology applications.

[0031] Trapped atomic ions are one of the most advanced qubit platforms, with atomic clock precision and the ability to perform gates in fully connected and reconfigurable qubit networks. The high connectivity between trapped ion qubits is mediated by the optical force of their collective motion. Trapped ions can be scaled in a modular manner by shuttling between independent ion chains in complex multi-electrode trap structures, or by photonic coupling on or off-chip. In a single large ion chain, gates can be achieved by appropriately shaping the laser pulses that drive selected ions within the chain. Here, the target qubits are entangled through Coulomb coupled motion, and the laser pulses are modulated so that the motion pattern is disentangled from the qubit at the end of the operation. Performing multiple parallel gates in this way requires the added complexity of the pulse shape to not only disentangle the motion, but also to entangle the target qubits only as needed. This type of parallel gate can be achieved with trapped ion technology by designing appropriate optical pulses using nonlinear optimization techniques.

[0032] Figure 1 A partial view of a vacuum chamber 100 is shown, which houses electrodes for trapping atomic ions in a chain or linear crystal 110 using an ion trap (e.g., a linear trap). Figure 1 In the example shown, the vacuum chamber in the quantum system (see for example Fig.10 ) includes electrodes for capturing a plurality of atomic ytterbium ions (e.g., 171Yb+ ions) that are confined in a linear crystal 110 and laser-cooled to near-stationary state. The number of captured atomic ions can be configurable, and in some cases, up to 100 or more atomic ions can be captured. The atoms are irradiated with laser radiation tuned to the 171Yb+ resonance, and the fluorescence of the atomic ions is imaged onto a camera. In this example, the atomic ions are spaced apart from each other by a distance 115 of about 5 micrometers (μm), as shown by the fluorescence. The separation of the atomic ions is determined by a balance between external restraining forces and Coulomb repulsion.

[0033] Strong fluorescence from a single trapped atomic ion relies on efficient recycling of photons, so the atomic structure of the ion must have strong closed optical transitions that allow laser cooling motion, qubit state initialization, and efficient qubit readout. This can rule out many atomic ion species, except for simple atomic ions with a single external electron, such as alkaline earth metals (Be+, Mg+, Ca+, Sr+, Ba+) and specific transition metals (Zn+, Hg+, Cd+, and Yb+). In these atomic ions, the qubit can be represented by two stable electronic energy levels, typically characterized by having two states |↑> and |↓>, or equivalently |1> and |0>.

[0034] In the present disclosure, a chain of five (5) atoms of Yb+ ions is used (see e.g. Figure 1 A chain or linear crystal 110 in FIG. 1 illustrates aspects of quantum gate parallelism, with resonant laser radiation used to laser cool, initialize, and measure qubits. It should be understood that the chain of trapped ions can be larger than five atomic ions. Coherent quantum gate operations are achieved by applying counter-propagating Raman beams from a single-mode locked laser that form beat notes near the qubit difference frequency. Single-qubit gates are generated by tuning the Raman beat note to the qubit frequency splitting ω0 and driving a resonant Rabi rotation (R(θ,φ) gate) with a defined phase and duration. Two-qubit (XX(χ)) gates are achieved by irradiating two ions at the beat note frequency near the motional sideband, generating an effective Ising interaction between the ions through transient entanglement of the motional modes. Similar operations can be used to generate gates between more than two qubits, but a pair is used in this example because two qubits are sufficient for universal logic. The pulse shaping scheme provides high-fidelity entangled gates on any ion pair. Although ion qubit systems have long coherence times relative to gate times, their coherence times are always limited, so parallel operations need to be used.

[0035] To perform parallel entanglement operations involving M independent qubit pairs in a chain of N ≥ 2M ions, k Next, a shaped qubit state-dependent force is applied to the involved ions using a two-color beat note pair with ω0±μ, yielding the evolution operator:

[0036]

[0037] where τ is the gate time. The first operator describes the state-dependent displacement of each mode k in the phase space, where

[0038]

[0039] And the cumulative displacement value

[0040]

[0041] here, and is a raising and lowering operator modulo k, η i,k is the Ramm–Dick parameter coupling qubit i to module k, and Ω i (t) is the Rabi frequency of the ith ion, proportional to the amplitude modulated laser intensity striking the ion. In some embodiments, frequency or phase modulation of the laser pulses may be used in conjunction with or in lieu of amplitude modulation. The second operator in equation (1) entangles qubits i and j and is given by:

[0042]

[0043] At the end of the gate operation, the 2M cumulative displacement values ​​involving the ion and the N modes in equation (3) should vanish so that all mode trajectories are closed in phase space and there is no remaining entanglement of the qubit motions. For each of the M desired entangled pairs, χ = π / 4 is necessary for maximum entanglement (or other non-zero values ​​of partial entanglement), and for other crosstalk pairs, χ = 0. This yields the value Ω for designing an appropriate pulse sequence i (t) to realize a total of M parallel entanglement gates To provide optimal control during gating and satisfy these constraints, we divide the laser pulse at ion i into S segments of equal duration τ / S and vary the amplitude in each segment as an independent variable. To achieve independent XX gates, independent signals are implemented on the M ion pairs to be entangled, which is necessary to provide sufficient control to entangle only the desired ion pairs simultaneously.

[0044] Although the 2MN motion pattern constraints (see, for example, Eq. (3)) are linear, The entanglement constraints (see, for example, Eq. (4)) are quadratic. Finding an impulsive solution to this non-convex quadratically constrained quadratic program (QCQP) is generally an NP-hard problem. Since analytical methods are intractable, optimization techniques can be used to find solutions that meet the constraints as closely as possible. The constraint problem setting and derivation of the fidelity of the XX gate operation is a function of the control parameters described above.

[0045] As described above, an illustrative embodiment of the parallel gate is an embodiment that can be implemented for two (2) independent ion pairs in a 5-ion chain. In one method, the optimization scheme uses the built-in MATLAB unconstrained multivariable optimization function "fminunc", where the objective function includes the above constraints on the α and χ parameters, as well as a term that minimizes power. It can be implemented for τ 门 = 250 μs gate time to compute the sequence, which is comparable to standard multi-qubit XX gates that have been used in other similar experiments, and can be computed for a range of detunings μ. This approach can yield a selection of solutions that can be tested on an experimental setup, where the solution that produces the highest quality gate using the least amount of power can be chosen.

[0046] The experimental setup is a five-capture 171 Yb + The qubits are designated as linear chains of ions that are laser-cooled to near their ground state. 2 S 1 / 2 Manifold |0>≡|F=0;m F =0> and |1〉≡|F=1; m F=0〉hyperfine split electronic states, which are first-order magnetic field-insensitive clock states, split at 12.642821 GHz. Coherent operations are performed by counter-propagating Raman beams from a single 355nm mode-locked laser. The first Raman beam is a global beam applied to the entire chain, while the second Raman beam is split into individual addressing beams to target each ion qubit. In addition, a multi-channel arbitrary waveform generator (AWG) provides individual RF control signals for individual addressing beams for each ion, providing individual phase, frequency, and amplitude control required to perform independent multi-qubit operations in parallel. That is, each ion is individually addressed using the addressing beam for that ion, and the characteristics of the addressing beam (e.g., phase, frequency, amplitude) are controlled to provide an appropriate pulse sequence for that ion. The qubit is initialized to the |0〉 state using optical pumping and read out by individual channels of a multi-channel photomultiplier tube (PMT) array using state-dependent fluorescence.

[0047] From the five trapped 171Yb+ ions, we found six experimental gates for ion pair combinations: (1, 4) and (2, 5); (1, 2) and (3, 4); (1, 4) and (2, 4); (1, 3) and (2, 5); (1, 2) and (4, 5). Figure 2A and Figure 2B The pulse sequences applied to each entangled pair to construct a set of parallel multi-qubit gates on ions (1, 4) and (2, 5) are shown in Figures 200a and 200b, respectively, along with the trajectories in the phase space of each mode pair interaction. The five transverse motion modes in these five ion chains have sideband frequencies ν x ={3.045, 3.027, 3.005, 2.978, 2.946} MHz, where mode 1 is the 3.045 MHz common mode. The phase space (P, X) trace (to the right of the pulse train) shows that the mode interactions closest to the chosen detuning exhibit the largest shifts and contribute the most to the final spin-spin entanglement by closing more phase space.

[0048] Negative amplitude pulses are achieved by inverting the phase of the control signal. This ability is useful because changing the control signal phase allows the entangled pair to continue accumulating entanglement while canceling the accumulated entanglement with the crosstalk pair. To this end, the initial guess used for the gate in the optimization protocol is that one pair will have all positive amplitude shapes, while the other pair will see positive amplitudes for the first half of the gate and negative amplitudes for the second half. Figure 2A and Figure 2B The pulse shape in provides a good example. Other pulse solutions (i.e., other pulse shapes and phase space trajectories) may have similar patterns with some symmetry, increasing and decreasing segment amplitudes, and flipping the phase on a pair to eliminate crosstalk entanglement.

[0049] Further relative to Figure 2A and Figure 2B , the pulse shape solutions and theoretical phase space trajectories shown are for parallel XX gates on ions (1,4) and (2,5) with gate time 250 μs, detuning = 2.962 MHz, and theoretical fidelity 99.63%. As described above, these figures illustrate the pulse sequences generated and applied to each entangled pair to construct the corresponding gate, showing the relative Rabi frequency during each segment of the gate. The negative segments have antiphase. The figures also illustrate the trajectories of each mode ion interaction in phase space, plotting α during the gate process. i,k . Ions with the same pulse shape are seen to have symmetric pattern ion trajectories. The phase space trajectory starts as an open circle and follows the path to the end. The axes in the plot are the same size, so the relative engagement of each mode is shown.

[0050] In this disclosure, experimental results for implementing parallel multi-qubit entangled gates on several selections of ion pairs are described. The fidelity is computed by performing the parallel gate and then performing an analysis pulse, and the fidelity is then determined using the computed parity of the result with the bare parallel gate. The analysis pulse is a rotation of the SK1 composite pulse used to increase robustness against errors in the rotation angle. For each of the four ions involved in the operation (e.g., the two ions for two parallel multi-qubit gates), parity analysis can be performed on all 6 possible pairs in the group, allowing analysis of two entangled ion pairs as well as 4 crosstalk pairs. Figure 3A and Figure 3B Parity check curves are shown in diagrams 300a and 300b, respectively. Entanglement gate fidelity is typically between 96-99%, with crosstalk accounting for a minority.

[0051] More about Figure 3A and Figure 3B , describes the parity curves and fidelity of parallel XX gates on two example sets of ions (e.g., sets (1, 4) and (2, 5) and sets (1, 4) and (2, 3). Figure 3A In Figure 300a, the set of ions (1, 4) and (2, 5) produces 96.5(4)% and 97.8(3)% fidelity on their respective entangled pairs, with an average crosstalk error of 3.6(3)%, and corrected state preparation and measurement (SPAM) error of 3%. Figure 3B In the diagram 300b in FIG. 300b, the set including ions (1, 4) and (2, 3) produces 98.8(3)% and 99.0(3)% fidelity on the respective entangled pairs, with an average crosstalk error of 1.4(3)%, and corrected SPAM error of <1%. These plots show both individual data points and fitted lines for these data points. For example, in Figure 3AIn the diagram 300a in FIG. 3 , the data points / fit lines for (1,4) (“14 data / fit) and for (2,5) (“25 data / fit) show the highest parity, and Figure 3B In the graph 300b of FIG. 300 , the data points / fit lines for (1,4) (“14 data / fit) and for (2,3) (“23 data / fit) show the highest parity.

[0052] The errors given above are statistical. The data has been corrected for SPAM errors. The crosstalk error is calculated by fitting the crosstalk pair parity sweep to a sinusoid as if it were a normal parity flip-flop, calculating its fidelity as an entanglement gate, and subtracting a 25% base fidelity representing full statistical mixing. Any fidelity above this indicates correlation or a small amount of entanglement and is considered erroneous for these purposes. All crosstalk fidelities for all pairs are close to 25%, indicating that no crosstalk pair has demonstrable entanglement.

[0053] As an example application of parallel operations useful for error correction codes, a pair of CNOT gates can be performed in parallel on two pairs of ions. A CNOT gate sequence (e.g., a compiled version of an R gate and an XX gate) is performed simultaneously on ion pair (1, 4), where ion 1 is used as a control and ion 4 is used as a target, and a gate sequence is performed simultaneously on ion pair (2, 3), where ion 2 is used as a control and ion 3 is used as a target. Each component operation in a composite gate is performed in parallel, where each rotation (e.g., R) is performed simultaneously with the corresponding rotation on the other pair, and both XX gates are performed using parallel XX gates on ion pairs (1, 4) and (2, 3). Simultaneous CNOT gates are performed for each of the 16 possible bitwise inputs, and the padding data for the 16 possible bitwise outputs is repeated in parallel. Figure 4 As shown in graph 400 in FIG. 4 , the average process fidelity is 94.5(2)% and the average SPAM error is 5%.

[0054] In addition to the examples given above, similar techniques may be useful in other types of quantum circuits or simulations. For example, in modern classical computing, a full adder is a basic circuit that can be cascaded to add multi-bit numbers and can be found in processors as a component of the arithmetic logic unit (ALU) and performs low-level operations such as calculating register addresses. A quantum full adder requires four (4) qubits, the first three (3) for the inputs x, y, and the carry bit C. in , and the fourth is a qubit initialized to |0>. The four outputs consist of the following: the first input x, which is simply passed through; y′, which carries (additional CNOTs can be added to extract y if necessary); and S and the output carry C out , which together include the inThe 2-bit result of the summation is out is the most significant bit and therefore becomes the carry bit for the next adder in the cascade, and S is the least significant bit. The sum can also be written as And the output carry can also be written as Feynman first designed such a circuit using CNOT and Toffoli gates, such as Figure 5A As shown in diagram 500a in FIG, this would require twelve (12) XX gates to implement on an ion trap quantum computer. A more efficient circuit has been described which requires a maximum of six (6) multi-qubit interactions. This has the further advantage of reducing the gate depth to 4 if simultaneous multi-qubit operations are available, as Figure 5B As shown in the dashed outline of diagram 500b.

[0055] exist Figure 7 Another embodiment of a full adder using 2 different parallel XX gate configurations and rotation and additional XX gates is shown in diagram 700 in FIG. Figure 5B The representation of the full adder in Figure 500b and the CNOT gate, C(V) gate and The parallel gates require different amounts of entanglement, which is equivalent to implementing a fully entangled The door and a partially entangled gates. This is achieved experimentally by adjusting the optical power supplied to each gate independently. Input x, y, C in and |0> are mapped to qubits (1, 2, 4, 5) respectively. Figure 6 Graph 600 in shows the resulting data for implementing this algorithm, with all 8 possible bitwise inputs on 3 input qubits, and showing the population of all 16 possible bitwise outputs on the 4 qubits used. This data yields an average process fidelity of 83.3(3)%, and corrects for an average SPAM error of 3%.

[0056] In conjunction with the examples described herein, computing or measuring the fidelity of a multi-qubit XX(χ) entangled gate can be performed by scanning the global This is accomplished by rotating the phase φ and calculating the parity at each point of the scan. Furthermore, the parallel gates can be calibrated independently of each other by adjusting the scaling factor that controls the total power on the gate without modifying the pulse shape. Adjusting the scaling factor that controls the power on a single ion will only affect the gate in which it participates by changing the total amount of entanglement, without any noticeable adverse effects on the gate quality. Additionally, since the XX gates in the parallelization scheme have independent calibrations, the χ parameters of the two XX gates can be independent. The continuous variable parameter χ is directly related to the amount of entanglement generated between the two qubits, and this parameter can be adjusted in experiments or implementations by scaling the power of the entire gate. Thus, as described above, two XX gates with different degrees of entanglement can be implemented simultaneously, which may prove useful in certain applications.

[0057] Faster multi-qubit gates can be done with more optical power, but this speedup is limited by the sideband resolution, a limitation that gets worse as processor size grows due to spectral crowding. Parallel multi-qubit operations are a tool to speed up computation that avoids this problem. Regarding optical power requirements, while the gate time τ of running 2 XX gates in parallel is 门 = 250μs is comparable to the gate time of a single XX gate (and therefore half the time required to execute two XX gates in series), but the parallel gate scheme requires more optical power. The Rabi frequency Ω is proportional to the square root of the beam intensity I, where I0 and I1 are the beam intensities of the individual beams and the global beam. We can therefore calculate the ratio of the power required for gates executed in parallel to the power required for a single XX gate on the same ion, R ||

[0058]

[0059] Since intensity is power per unit area, and the beam size does not change, this cancels out. The power ratios for each gate are shown in Table I. While some gates require more power (e.g., high quality and low power solutions to (1,2), (3,4) are a challenge), most gates executed in parallel require 2 to 4 times the power of their single-executed counterparts. However, a full accounting of the power requirements for this experiment must also take into account the power wasted by unused beams and the total time required to execute an equivalent operating beam at any time and be discarded after the AOM when not in use, with any unirradiated ions corresponding to the wasted power of a single beam. Running 2 XX gates in parallel takes τ 门 = 250 μs and use each beam with power P to illuminate 4 ions, but using independent XX gates to perform the same 2 gates in series requires time 2τ 门And using 4 beams with power P / 4 to P / 2 each to illuminate 2 ions, 2 beams are wasted. This leaves the choice of using twice the power (or more) in half the time and half the power in twice the time, the result depending on the gate in question. Parallel gates are therefore very useful when the laser power is available and the benefit of parallelism outweighs the time.

[0060]

[0061] Table I shows the power ratio R of each pair of parallel XX gates implemented. || to compare the power required to execute each component XX and its corresponding independent multi-qubit XX gate.

[0062] The control scheme for parallel multi-qubit entanglement gates for ions presented in this paper also proposes a method for performing multi-qubit entanglement in a single operation. Of particular interest is the creation of Greenberger-Horne-Zeilinger (GHZ) states. GHZ states are a specific type of entangled quantum state that involves at least three subsystems (particles). In other words, GHZ states are a class of non-doubly separable maximally entangled multi-qubit states.

[0063] The same optimization methods that produce pulse shapes for parallel multi-qubit entanglement gates can be used to produce GHZ states when applied to ions. However, unlike the parallel gates, it may be necessary to allow independent pulse shapes on all four ions, rather than solving for pairwise solutions; this would provide more free parameters. Additional challenges may include finding efficient calibration techniques when implementing such a gate in an experiment, since there will be six interactions that all need to be at the same strength, but only four control signals. Current methods of calibrating multi-qubit gates by adjusting the total power of the pulse shape imposed by the control signals may have limited applicability to GHZ states and techniques involving more degrees of freedom that may require, such as independently adjusting the power of different segments of the pulse shape on each ion.

[0064] The benefit of implementing GHZ states with fewer gates is significant as it will greatly reduce the circuit depth of several important algorithms. While the use of axial modes for multi-qubit GHZ states has been described, the scheme described in this paper represents a new approach for radial mode interactions. In the case where only multi-qubit gates are available, establishing a GHZ state of size N requires O(N) multi-qubit gates. However, in the case where parallel multi-qubit gates are available, the gate depth required to establish the GHZ state is reduced to O(log(N)). This can be implemented using a binary tree algorithm by dividing all qubits into pairs and entangling the pairs in parallel, then entangling the pairs within the pairs, and so on, until all are entangled. A single operation GHZ state will bring this circuit depth down to unity. The single operation GHZ state construction will greatly improve the efficiency of a variety of algorithms. For example, an arbitrary stabilizer circuit requires The creation of a single-operation GHZ state will also benefit applications such as quantum secret sharing, Toffoli-N gates, quantum Fourier transforms, and quantum Fourier adder circuits.

[0065] Another aspect of the disclosure is the scalability of the techniques described herein. For example, the scaling prospects of synchronous gates are polynomial or better in the number of constraints to consider when computing the optimal solution. Two parallel XX gates in a chain of N ions require 4N+6 ~ O(N) constraints, so the problem grows linearly in N. Entangling more pairs at a time grows quadratically: Entangling M pairs involves the interaction of 2M ions, yielding the number of spin-spin interactions we must control as

[0066]

[0067] And the number of spin-motion interactions is the number of ions times the number of modes, 2MN. Therefore, scaling both the number of entangled pairs M and the number of ions in the chain N gives a total problem growth rate of

[0068] 2MN+2M 2 -M~O(M 2 +MN). (7)

[0069] On very long chains, not all ion-ion connections are directly available, which reduces the number of quadratic constraints on crosstalk pairs that must be considered, suggesting that this is an upper limit on scaling.

[0070] To improve the techniques described in this paper, it may be desirable to increase the fidelity of the solution. Relaxing the constraints on the required power can compute higher fidelity solutions, although increasing the power of the experiment will exacerbate errors caused by Raman beam noise. One improvement may involve determining whether the constraint matrix based on the piecewise amplitudes of the entangled pairs can be modified to be positive semidefinite or negative semidefinite, as convex QCQPs are easy to solve using semidefinite programming techniques and may allow higher fidelity solutions. However, these are overhead issues. Once a high-quality gate solution is achieved in the experiment, no further computation is required and only a single calibration is required to compensate for Rabi frequency drift.

[0071] Now refer to Figure 8 , Figure 8 An example computer device 800 is shown in accordance with various aspects of the present disclosure. For example, computer device 800 may represent a single computing device, multiple computing devices, or a distributed computing system. Computer device 800 may be configured as a quantum computer, a classical computer, or a combination of quantum and classical computing functions. For example, computer device 800 may be used to process information to generate or implement the various techniques described herein for implementing parallel multi-qubit operations in different quantum circuits, algorithms, experiments, or simulations, and any related features, including, for example, performing multi-qubit entanglement (e.g., GHZ states) in a single operation. Computer device 800 may be used as a quantum computer or QIP system, such as, but not limited to, a trapped ion quantum computer or system. In Fig.10 A general example of a QIP system that can be used to implement aspects of computer device 800 is illustrated in the example shown in FIG.

[0072] In one example, the computer device 800 may include a processor 810 for performing processing functions associated with one or more of the features described herein. For example, the processor 810 may be used to control the operation of the computer device 800 to implement, perform and / or execute parallel multi-qubit operations in different quantum circuits, algorithms, experiments or simulations, and any related features, for example, including executing multi-qubit entanglement (e.g., GHZ states) in a single operation. The processor 810 may include a single or multiple groups of processors or a multi-core processor. In addition, the processor 810 may be implemented as an integrated processing system and / or a distributed processing system. The processor 810 may include a central processing unit (CPU), a quantum processing unit (QPU), a graphics processing unit (GPU), or a combination of these types of processors.

[0073] In one example, the computer device 800 may include a memory 820 for storing instructions executable by the processor 810 for performing the functions described herein. In one embodiment, for example, the memory 820 may correspond to a computer-readable storage medium storing code or instructions to perform one or more of the functions or operations described herein. In one example, the memory 820 may include instructions for performing the following combined Fig. 9 Instructions for various aspects of the described method 900. In one example, the memory 820 can be used to store information about pulse sequences used to generate different sets of parallel multi-qubit gates.

[0074] In addition, the computer device 800 may include a communication component 830 for establishing and maintaining communications with one or more parties using hardware, software, and services as described herein. The communication component 830 may host communications between components on the computer device 800 and between the computer device 800 and external devices (e.g., devices located on a communication network and / or devices connected in series or locally to the computer device 800). For example, the communication component 830 may include one or more buses and may further include a transmit chain component and a receive chain component associated with a transmitter and a receiver, respectively, which are operable to communicate with external devices.

[0075] In addition, computer device 800 may include data repository 840, which may be any suitable combination of hardware and / or software that provides a large-capacity repository of information, databases, and programs employed by the embodiments described herein. For example, data repository 840 may be a data repository for operating system 860 (e.g., a classical OS or a quantum OS). In one embodiment, data repository 840 may include memory 820.

[0076] The computer device 800 may also include a user interface component 850, which is operable to receive input from a user of the computer device 800, and is also operable to generate output for presenting to the user or providing to different systems (directly or indirectly). The user interface component 850 may include one or more input devices, including but not limited to a keyboard, numeric keys, mouse, touch-sensitive display, digitizer, navigation keys, function keys, microphone, voice recognition component, any other mechanism that can receive input from the user, or any combination thereof. In addition, the user interface component 850 may include one or more output devices, including but not limited to a display, a speaker, a tactile feedback mechanism, a printer, any other mechanism that can present output to the user, or any combination thereof.

[0077] In one embodiment, the user interface component 850 may send and / or receive messages corresponding to the operation of the operating system 860. In addition, the processor 810 may execute the operating system 860 and / or applications or programs (e.g., programs, algorithms, simulations, experiments using quantum gates), and the memory 820 or data repository 840 may store them.

[0078] When the computer device 800 is implemented as part of a cloud-based infrastructure solution, the user interface component 850 may be used to allow a user of the cloud-based infrastructure solution to interact with the computer device 800 remotely.

[0079] Fig. 9 8 is a flow chart illustrating an example of a method 900 for performing quantum operations in a trapped ion quantum system according to aspects of the present disclosure. In one aspect, the method 900 can be performed in a trapped ion quantum system such as the computer device 800 described above, where, for example, the processor 810, the memory 820, the data repository 840, and / or the operating system 860 can be used to perform the functions of the method 900. The trapped ion quantum system used to perform the method 900 can also be a QIP system, as described below in conjunction with Fig.10 Describe the QIP system.

[0080] At 910, method 900 includes implementing at least two parallel gates of a quantum circuit, each of the at least two parallel gates being a multi-qubit gate, each of the at least two parallel gates being implemented using different ion pairs of a plurality of ions in an ion trap, and the plurality of ions comprising four or more ions.

[0081] At 920, method 900 includes simultaneously performing an operation on the at least two parallel gates as part of a quantum operation.

[0082] In another aspect of method 900, the at least two parallel gates include a first gate and a second gate, the first gate is implemented using a first ion pair from the plurality of ions, and the second gate is implemented using a second ion pair from the plurality of ions, the first ion pair and the second ion pair are entangled pairs, and any remaining ion pairs from the plurality of ions in the ion trap are not entangled pairs. The ions from the first ion pair are located at any two positions in the ion trap, and the ions from the second ion pair are located at any two positions from the remaining positions in the ion trap.

[0083] In another aspect of method 900, implementing the at least two parallel gates of the quantum circuit includes generating a sequence of optical pulses that is applied to the first ion pair to construct the first gate and is applied to the second ion pair to construct the second gate in parallel with the first gate. In addition, the generation of the sequence of optical pulses includes retrieving stored information to generate a sequence of optical pulses for constructing the first gate using the first ion pair and constructing the second gate using the second ion pair. In addition, the generation of the sequence of optical pulses includes: generating a first pulse sequence to construct the first gate, the first pulse sequence having a plurality of segments for the first gate, each segment of the first pulse sequence corresponding to a relative frequency during the segment of the first gate, and the plurality of segments of the first pulse sequence having equal durations; and generating a second pulse sequence to construct the second gate, the second pulse sequence having a plurality of segments for the second gate, each segment of the second pulse sequence corresponding to a relative frequency during the segment of the second gate, and the plurality of segments of the second pulse sequence having equal durations.

[0084] In another aspect of method 900, the at least two parallel gates are both XX gates.

[0085] In another aspect of method 900, the at least two parallel gates are both CNOT gates.

[0086] In another aspect of method 900, the at least two parallel gates have different amounts of entanglement. For example, the at least two parallel gates include a first gate and a second gate, and the first gate is a fully entangled gate and the second gate is a partially entangled gate. In another example, the at least two parallel gates include fully entangled Gates and Partial Entanglement Door.

[0087] In another aspect of method 900, the quantum circuit may be any of the various quantum circuits or simulations described herein, including, for example, a quantum full adder circuit.

[0088] In another aspect of method 900, the at least two parallel gates include a first 2-qubit gate and a second 2-qubit gate, the first gate is implemented using a first ion pair from among the plurality of ions in the ion trap, and the second gate is implemented using a second ion pair from among the plurality of ions in the ion trap, the first ion pair and the second ion pair are entangled pairs, and any remaining ion pairs of the plurality of ions in the ion trap are not entangled pairs. The ions in the first ion pair may be located at any two positions in the ion trap, and the ions in the second ion pair may be located at any two positions in the remaining positions in the ion trap.

[0089] It should be understood that method 900 can be applied to multiple parallel gates, each of which is a multi-qubit gate. For example, two or more parallel gates (e.g., 2 parallel gates, 3 parallel gates, 4 parallel gates, etc.) can be implemented, each of which is a multi-qubit gate (e.g., 2-qubit gate, 3-qubit gate, 4-qubit gate, etc.).

[0090] It should also be understood that the various aspects described above in conjunction with method 900 can be combined so that two or more of these aspects represent different embodiments supported by the present disclosure. For example, two or more aspects of method 900 can be combined to produce an embodiment associated with method 900.

[0091] Fig.10 1 is a block diagram illustrating an example of a QIP system 1000 according to aspects of the present disclosure. The QIP system 1000 may also be referred to as a quantum computing system, a computer device, etc. In one aspect, the QIP system 1000 may correspond to Figure 8 Portions of a quantum computer implementation of computing device 800 in FIG.

[0092] The QIP system 1000 may include a source 1060 that provides atomic species to a chamber 1050 having an ion trap 1070 that captures the atomic species ionized by the optical controller 1020. The chamber 1050 may correspond to the above-described Figure 1 The vacuum chamber 100 and the ion trap 1070 can be used to maintain the ion chain, for example, the above is also combined with Figure 1 The chain or linear crystal 110 described. The light source 1030 in the optical controller 1020 may include one or more laser sources that can be used to: ionize atomic species; control atomic ions (e.g., control phase); cause atomic ions to emit fluorescence, which can be monitored and tracked by an image processing algorithm running in an imaging system 1040 in the optical controller 1020; and generate a sequence of light pulses that is used to implement parallel multi-qubit gates and related operations. The imaging system 1040 may include a high-resolution imager (e.g., a CCD camera) for monitoring the atomic ions as they are provided to the ion trap 1070 (e.g., for counting) or after the atomic ions are provided to the ion trap 1070 (e.g., for monitoring the atomic ion state). In one aspect, the imaging system 1040 may be implemented separately from the optical controller 1020, however, the use of image processing algorithms to detect, identify, and label atomic ions using fluorescence may require cooperation with the optical controller 1020.

[0093] The QIP system 1000 may also include an algorithm component 1010 that may operate with other portions of the QIP system 1000 (not shown) to implement, perform, and / or execute quantum operations, including quantum circuits, quantum algorithms, and quantum simulations as described above. Thus, the algorithm component 1010 may provide instructions to various components of the QIP system 1000 (e.g., to the optical controller 1020) to enable implementation, performance, and / or execution of quantum operations, particularly for quantum operations involving parallel multi-qubit gates and related operations.

[0094] In the example of QIP system 1000 (which may also be referred to as a trapped ion quantum system), algorithm component 1010 may be configured to implement at least two parallel gates of a quantum circuit, each of the at least two parallel gates being a multi-qubit gate, each of the at least two parallel gates being implemented using a different ion pair of a plurality of ions in ion trap 1070, and the plurality of ions comprising four or more ions. In ion trap 1070, operations are performed simultaneously on the at least two parallel gates as part of a quantum operation.

[0095] In addition, the algorithm component 1010 is also configured to provide instructions to the optical controller 1020 so that the optical controller 1020 generates an optical pulse sequence based on the instructions, and the optical pulse sequence is applied to the different ion pairs to construct the at least two parallel gates. The optical controller 1020 is also configured to retrieve the stored information based on the instructions to generate the optical pulse sequence. The optical controller 1020 is configured to: generate a first pulse sequence to construct a first gate of the at least two parallel gates, the first pulse sequence having a plurality of segments for the first gate, each segment of the first pulse sequence corresponding to a relative frequency during the segment of the first gate, and the plurality of segments of the first pulse sequence having equal durations; and generate a second pulse sequence to construct a second gate of the at least two parallel gates, the second pulse sequence having a plurality of segments for the second gate, each segment of the second pulse sequence corresponding to a relative frequency during the segment of the second gate, and the plurality of segments of the second pulse sequence having equal durations.

[0096] It should be understood that the examples associated with the QIP system 1000 are provided by way of illustration and not limitation. In different implementations or embodiments, features or functions of one component or subcomponent of the QIP system 1000 may be combined with other components or subcomponents. In addition, detailed features or functions supported by the QIP system 1000 that are not explicitly associated with a particular component or subcomponent may be appropriately performed by one of the existing components or subcomponents. For example, aspects related to quantum circuit and gate selection, implementation, and execution control may be associated with the algorithm component 1010, aspects related to optical operations may be associated with the optical controller 1020, and aspects related to the interaction of trapped ions may be associated with the ion trap 1070.

[0097] It should be understood that QIP system 1000 can be applied to multiple parallel gates, each of which is a multi-qubit gate. For example, two or more parallel gates (e.g., 2 parallel gates, 3 parallel gates, 4 parallel gates, etc.) can be implemented, each of which is a multi-qubit gate (e.g., 2-qubit gate, 3-qubit gate, 4-qubit gate, etc.).

[0098] It should also be understood that the various aspects described above in conjunction with QIP system 1000 can be combined such that two or more of these aspects represent different embodiments supported by the present disclosure. For example, two or more aspects of QIP system 1000 can be combined to produce an embodiment associated with QIP system 1000.

[0099] Although the present disclosure has been provided according to the illustrated embodiments, it will be readily appreciated by those skilled in the art that variations of the embodiments may exist and that these variations will be within the scope of the present disclosure. Therefore, many modifications may be made by those skilled in the art without departing from the scope of the appended claims.

Claims

1. A method for performing quantum operations in a trapped ion quantum system, the method comprising: implementing two parallel gates of a quantum circuit, each of the two parallel gates being a 2-qubit gate, each of the two parallel gates being implemented using a different set of ions in a plurality of ions in an ion trap, the two parallel gates comprising a first 2-qubit gate implemented using a first set of ions and a second 2-qubit gate implemented using a second set of ions, wherein the first set of ions and the second set of ions are entangled sets, and any remaining set of ions in the plurality of ions in the ion trap is not an entangled set, wherein implementing the two parallel gates comprises generating a first light pulse sequence and applying a first light beam based on the first light pulse sequence to the first set of ions to implement the first gate, wherein a solution of the first light pulse sequence is based on a quadratic entanglement constraint and has a plurality of segments, each segment of the first light pulse sequence corresponds to a relative frequency during the segment of the first gate, and the segments of the first light pulse sequence have equal durations, and wherein implementing the two parallel gates comprises generating a second light pulse sequence and applying a second light beam based on the second light pulse sequence to the second set of ions to implement the second gate, wherein a solution of the second light pulse sequence is based on a quadratic entanglement constraint and has a plurality of segments for the second gate, each segment of the second light pulse sequence corresponds to a relative frequency during the segment of the second gate, and the segments of the second light pulse sequence have equal duration; as well as As part of the quantum operation, an operation is performed simultaneously on the two parallel gates.

2. The method according to claim 1, wherein: The ions in the first set of ions are located anywhere in the ion trap, and Ions in the second set of ions are located in any remaining locations in the ion trap.

3. The method according to claim 1, wherein: The generating of the first and second sequences of light pulses includes retrieving stored information to generate the first and second sequences of light pulses.

4. The method according to claim 1, wherein: The two parallel gates are both XX gates.

5. The method according to claim 1, wherein: The two parallel gates are both CNOT gates.

6. The method according to claim 1, wherein: The two parallel gates have different amounts of entanglement.

7. The method according to claim 6, wherein: The first gate is a fully entangled gate and the second gate is a partially entangled gate.

8. The method according to claim 6, wherein: The two parallel gates include fully entangled Gates and Partial Entanglement Door.

9. The method according to claim 1, wherein: The quantum circuit is a quantum full adder circuit.

10. A trapped ion quantum system configured to perform quantum operations, the trapped ion quantum system comprising: an algorithm component configured to implement two parallel gates of a quantum circuit, each of the two parallel gates being a 2-qubit gate, each of the two parallel gates being implemented using a different set of ions from a plurality of ions in an ion trap, the two parallel gates comprising a first 2-qubit gate implemented using a first set of ions and a second 2-qubit gate implemented using a second set of ions, wherein the first set of ions and the second set of ions are entangled sets, and any remaining set of ions from the plurality of ions in the ion trap is not an entangled set; an optical controller, wherein the algorithm component is configured to provide instructions to the optical controller, and the optical controller is configured to generate a first light pulse sequence based on the instructions and apply a first light beam based on the first light pulse sequence to the first set of ions to implement the first gate, wherein a solution of the first light pulse sequence is based on a quadratic entanglement constraint and has a plurality of segments, each segment of the first light pulse sequence corresponds to a relative frequency during the segment of the first gate, and the segments of the first light pulse sequence have equal durations, wherein the optical controller is further configured to generate a second light pulse sequence based on the instruction and apply a second light beam based on the second light pulse sequence to the second set of ions to implement the second gate, wherein a solution of the second light pulse sequence is based on a quadratic entanglement constraint and has a plurality of segments for the second gate, each segment of the second light pulse sequence corresponds to a relative frequency during the segment of the second gate, and the segments of the second light pulse sequence have equal duration; and The ion trap, wherein, as part of the quantum operation, an operation is performed simultaneously on the two parallel gates.

11. The trapped ion quantum system of claim 10, wherein: The ions in the first set of ions are located anywhere in the ion trap, and Ions in the second set of ions are located in any remaining locations in the ion trap.

12. The trapped ion quantum system according to claim 10, wherein: The optical controller configured to generate the first and second trains of light pulses is further configured to retrieve the stored information based on the instructions to generate the first and second trains of light pulses.

13. The trapped ion quantum system according to claim 10, wherein: The two parallel gates are both XX gates.

14. The trapped ion quantum system according to claim 10, wherein: The two parallel gates are both CNOT gates.

15. The trapped ion quantum system according to claim 10, wherein: The two parallel gates have different amounts of entanglement.

16. The trapped ion quantum system of claim 15, wherein: The first gate is a fully entangled gate and the second gate is a partially entangled gate.

17. The trapped ion quantum system according to claim 15, wherein: The two parallel gates include fully entangled Gates and Partial Entanglement Door.

18. The trapped ion quantum system according to claim 10, wherein: The quantum circuit is a quantum full adder circuit.

19. A computer-readable storage medium storing code having instructions executable by a processor for performing quantum operations in a trapped ion quantum system, the computer-readable storage medium comprising: code for implementing two parallel gates of a quantum circuit, each of the two parallel gates being a 2-qubit gate, each of the two parallel gates being implemented using a different set of ions from a plurality of ions in an ion trap, the two parallel gates comprising a first 2-qubit gate implemented using a first set of ions and a second 2-qubit gate implemented using a second set of ions, wherein the first set of ions and the second set of ions are entangled sets, and any remaining set of ions from the plurality of ions in the ion trap is not an entangled set; code for generating a first train of light pulses and applying a first light beam based on the first train of light pulses to the first set of ions to implement the first gate, wherein a solution of the first train of light pulses is based on a quadratic entanglement constraint and has a plurality of segments, each segment of the first train of light pulses corresponds to a relative frequency during the segment of the first gate, and the segments of the first train of light pulses have equal duration; code for generating a second train of light pulses and applying a second light beam based on the second train of light pulses to the second set of ions to implement the second gate, wherein a solution of the second train of light pulses is based on a quadratic entanglement constraint and has a plurality of segments for the second gate, each segment of the second train of light pulses corresponding to a relative frequency during the segment of the second gate, and the segments of the second train of light pulses have equal duration; and Code for simultaneously performing an operation on the two parallel gates as part of the quantum operation.

20. The computer-readable storage medium of claim 19, wherein: The ions in the first set of ions are located anywhere in the ion trap, and Ions in the second set of ions are located in any remaining locations in the ion trap.

Images