Patent Description:
<NPL>-<NUM> teaches different ways of implementing a CNOT in a qubit lattice all following the principle of a Z⊗Z parity measurement between the control and an ancilla initialized in the X eigenstate |+〉, a subsequent X⊗X parity measurement between ancilla and target, and a final Z basis readout of the ancilla qubit.

<NPL> relates to resource estimation of a quantum algorithm on a topological quantum computer, for the purpose of resource optimization.

<NPL>, relates to analysis of physical parameters of a topological quantum memory in terms or errors rates.

According to one aspect of the present disclosure, a computing system is provided, in accordance with independent claim <NUM>. According to another aspect of the present disclosure, a method is provided, in accordance with independent claim <NUM>.

Measurement-only topological quantum computation is one approach to topological quantum computation that is well-suited to implementation using Majorana zero modes (MZMs). Measurement-only topological quantum computation allows computations to be performed without physically moving the MZMs, which are typically bound to macroscopic defects (such as the ends of wires, as discussed in further detail below) and may be difficult to move. Instead, braiding transformations may be performed through a series of (potentially non-local) measurements on sets of MZMs involving the MZMs that encode the computational state that is to be manipulated and another set of MZMs that serve as ancillary degrees of freedom. The MZMs may, in some architectures, be coupled to quantum dots, thus allowing the states of the MZMs to be measured by measuring the effects of the MZMs on the energy spectra of the quantum dots.

Using MZMs, a Clifford gate (for example, a Hadamard gate, a π/<NUM> phase gate, or a controlled not gate) may be constructed in a quantum computing device. In order to perform universal quantum computation, an additional gate such as a T-gate (a π/<NUM> phase gate) may additionally be implemented. The Clifford gate may be topologically protected such that perturbations to the quantum state are suppressed when the Clifford gate is implemented. The T-gate may not be topologically protected in such configurations.

Each Clifford gate implemented at the quantum computing device may be compiled from a sequence of measurements of the quantum state. According to previous approaches for constructing a Clifford gate from a sequence of measurements, a measurement sequence with minimal length is generated for each of the basic braiding transformations for each qubit. In such approaches, a minimal-length measurement sequence for a two-qubit entangling gate is then generated for each pair of qubits, and the resulting gate set is used as a generating gate set to synthesize any other Clifford gates. However, this approach may be inefficient, since there may exist shorter sequences of measurements that compile to the same gate.

In order to address the above inefficiency of existing methods for compiling Clifford gates in topological quantum computing devices, a computing system <NUM> is provided, as shown in the example embodiment of <FIG>. The computing system <NUM> may include a processor <NUM> and memory <NUM>, which may be operatively coupled. The processor <NUM> and memory <NUM> may be included in a classical computing device. The computing system <NUM> may further include other hardware components such as one or more input devices, one or more output devices, and/or one or more communication devices. In some embodiments, the functions of the computing system <NUM> may be distributed across a plurality of communicatively coupled computing devices.

As discussed in further detail below, the processor <NUM> may be configured to identify a plurality of measurement sequences <NUM> that implement a logic gate <NUM>. The logic gate <NUM> may be a one-qubit Clifford gate or a multi-qubit Clifford gate. In some embodiments, as shown in the example of <FIG>, the computing system <NUM> may include a topological quantum computing device <NUM> having a quantum state <NUM>. Each measurement sequence <NUM> of the plurality of measurement sequences <NUM> may include a plurality of measurements <NUM> of the quantum state <NUM> of the topological quantum computing device <NUM>. In such embodiments, the processor <NUM> may be operatively coupled to the topological quantum computing device <NUM> and may be configured to transmit one or more measurement sequences <NUM> to the topological quantum computing device <NUM> to implement the one or more logic gates <NUM>. In other embodiments, the processor <NUM> may be configured to identify the plurality of measurement sequences <NUM> for later use with a topological quantum computing device <NUM> when the computing system <NUM> does not include a topological quantum computing device <NUM>.

The processor <NUM> may be further configured to determine a respective estimated total resource cost <NUM> of each measurement sequence <NUM> of the plurality of measurement sequences <NUM>. The estimated total resource cost <NUM> of a measurement sequence <NUM> may, for example, indicate an amount of time or an amount of energy that implementing the logic gate <NUM> with the measurement sequence <NUM> is estimated to consume. Additionally or alternatively, the estimated total resource cost <NUM> may indicate an error rate of the measurement sequence <NUM>.

In some embodiments, for each measurement sequence <NUM> of the plurality of measurement sequences <NUM>, the processor <NUM> may be configured to determine the respective estimated total resource cost <NUM> at least in part by determining an estimated weighted resource cost <NUM> of each measurement <NUM> included in the measurement sequence <NUM>. For example, the estimated weighted resource cost <NUM> of each measurement <NUM> may indicate an error rate of the measurement <NUM>. The processor <NUM> may then be further configured to determine the estimated total resource cost <NUM> of the measurement sequence <NUM> based on the plurality of estimated weighted resource costs <NUM> of the individual measurements <NUM>. For example, the estimated total resource cost <NUM> may be the product of the estimated weighted resource costs <NUM>.

Once the respective estimated total resource costs <NUM> of the plurality of measurement sequences <NUM> have been determined, the processor <NUM> may be further configured to determine a first measurement sequence <NUM> that has a lowest estimated total resource cost <NUM> of the plurality of measurement sequences <NUM>. The processor <NUM> may, in some embodiments, be further configured to transmit the first measurement sequence <NUM> to the topological quantum computing device <NUM> so that the topological quantum computing device <NUM> may implement the logic gate <NUM> by applying the first measurement sequence <NUM> to the quantum state <NUM>.

The topological quantum computing device <NUM> may include one or more Majorana hexons, as shown in <FIG>. The example of <FIG> shows a first Majorana hexon 30A. In some embodiments, the topological quantum computing device <NUM> may further include a second Majorana hexon 30B. Each Majorana hexon may include six Majorana zero modes (MZMs). The first Majorana hexon 30A includes Majorana zero modes 32A, 32B, 32C, 32D, 32E, and 32F. The second Majorana hexon 30B includes Majorana zero modes <NUM>, <NUM>, 32I, 32J, <NUM>, and <NUM>. In some embodiments, the topological quantum computing device <NUM> may include more than two Majorana hexons.

Example structures that may be included in the topological quantum computing device <NUM> to implement one or more qubits are shown in <FIG> and <FIG>. The topological quantum computing device <NUM> may instantiate the plurality of MZMs <NUM> in a one-sided architecture or a two-sided architecture. <FIG> shows an example of a two-sided Majorana hexon architecture <NUM>. In the embodiment of <FIG>, the MZMs <NUM> included in each Majorana hexon <NUM> are coupled by a topological superconductor <NUM> and a superconductor <NUM>. Within each Majorana hexon <NUM>, the superconductor <NUM> is located between a first side 72A on which three MZMs <NUM> (labeled γ<NUM>, γ<NUM>, and γ<NUM>) are located and a second side 72B on which the other three MZMs <NUM> (labeled γ<NUM>, γ<NUM>, and γ<NUM>) are located. The two-sided Majorana hexon architecture <NUM> of <FIG> includes a plurality of Majorana hexons <NUM>, and further includes a semiconductor <NUM> that is located between adjacent Majorana hexons <NUM> and is coupled to the topological superconductor <NUM>. Each MZM <NUM> may form at a junction between the topological superconductor <NUM> and the semiconductor <NUM>. When the two-sided Majorana hexon architecture <NUM> of <FIG> is used, each Majorana hexon <NUM> may have a square lattice connectivity in which each Majorana hexon <NUM> may be entangled with its vertical and horizontal neighbors.

An example a one-sided Majorana hexon architecture <NUM> is shown in the example of <FIG>. In the one-sided Majorana hexon architecture <NUM>, each MZM <NUM> included in a Majorana hexon <NUM> is located on the same side of the superconductor <NUM>. As in the example embodiment of <FIG>, each MZM <NUM> may be formed at a junction between a topological superconductor <NUM> and a semiconductor <NUM>. <FIG> further shows a close-up view of a portion of the semiconductor <NUM> in the one-sided Majorana hexon architecture <NUM>. In the example of <FIG>, a plurality of quantum dots <NUM> are embedded within the semiconductor <NUM> proximate the MZMs <NUM>. In addition, the close-up view of the portion of the semiconductor <NUM> shows a plurality of cutter gates <NUM> located between the quantum dots <NUM> and the MZMs <NUM> and between pairs of quantum dots <NUM>. The cutter gates <NUM> and the quantum dots <NUM> are described in further detail below.

In the one-sided Majorana hexon architecture <NUM> of <FIG>, the plurality of Majorana hexons <NUM> may have an asymmetric square lattice connectivity in which the one-sided Majorana hexon architecture <NUM> does not have vertical reflection symmetry. Thus, two-qubit gates acting on horizontally displaced qubits may have different leftward and rightward operations. A two-qubit gate included in the two-sided Majorana hexon architecture <NUM> of <FIG> may also have different leftward and rightward operations if the labeling of the MZMs <NUM> is asymmetric under left-right reflection.

In the two-sided Majorana hexon architecture <NUM> and the one-sided Majorana hexon architecture <NUM>, each Majorana hexon <NUM> may be galvanically isolated from other Majorana hexons <NUM>. Thus, each Majorana hexon <NUM> may have a charging energy EC resulting from Coulomb interactions. The charging energy of each Majorana hexon <NUM> may reduce the probability of quasiparticle poisoning of that Majorana hexon <NUM>, since the probability for an electron to tunnel onto or off of the island is exponentially suppressed in the ratio of the charging energy EC to temperature, exp(-EC/kBT), where T is the temperature of the Majorana hexon <NUM> and kB is Boltzmann's constant.

Projective measurements of the joint fermionic parity of any two MZMs <NUM> may be performed by enabling weak coherent single-electron tunneling between the MZMs <NUM> included in the pair and the quantum dots <NUM> adjacent to those MZMs <NUM>. This coupling gives rise to a shift in the energy spectrum and charge occupation of the dot, which may then be measured. This measurement may be topologically protected in the sense that the operator that is being measured is known up to corrections that are exponentially small in the separation distance, through the superconductor <NUM> and the topological superconductor <NUM>, of MZMs <NUM> in the pair. However, measurement fidelity may be limited by the signal-to-noise ratio and decoherence of the qubit.

In each Majorana hexon <NUM>, a qubit may be formed from four MZMs <NUM>. The remaining two MZMs <NUM> included in the Majorana hexon <NUM> may be used as ancillary MZMs <NUM> when performing measurement-based topological operations, as discussed in further detail below. In other embodiments, additionally or alternatively to a Majorana hexon <NUM>, the topological quantum computing device may include two Majorana tetrons. <FIG> shows an example computing system <NUM> including a topological quantum computing device <NUM> with a quantum state <NUM>. The topological quantum computing device <NUM> of <FIG> includes two Majorana tetrons. In the embodiment of <FIG>, one Majorana tetron is a computational Majorana tetron 130A that encodes a computational qubit and the other Majorana tetron is an ancillary Majorana tetron 130B that encodes an ancillary qubit. The computational Majorana tetron 130A includes four Majorana zero modes 132A, 132B, 132C, and 132D. The ancillary Majorana tetron 130B includes Majorana zero modes 132E, 132F, <NUM>, and <NUM>. In other embodiments, the topological quantum computing device <NUM> may include some other number of Majorana tetrons.

In other embodiments not shown in the figures, the topological quantum computing device <NUM> may include one or more Majorana octons that each include eight MZMs <NUM>.

Some measurements <NUM> of the quantum state <NUM> may be more difficult to perform than other measurements <NUM>. As discussed above, the difficulty of performing a measurement <NUM> may be indicated by an estimated weighted resource cost <NUM> of that measurement <NUM>. In the two-sided Majorana hexon architecture <NUM> of <FIG>, some measurements <NUM> may be performed between MZMs <NUM> on the same side (left or right) of the Majorana hexon <NUM>. Other measurements may be performed between MZMs <NUM> on opposite sides of the Majorana hexon <NUM>. When the MZMs <NUM> have a greater separation distance, enabling coherent single-electron tunneling between these MZMs <NUM> and a common quantum dot <NUM> may be more difficult, since the distance between the MZMs <NUM> may exceed the phase coherence length of the semiconductor <NUM>. Thus, the estimated weighted resource cost <NUM> of a measurement <NUM> may increase with increasing distance between the MZMs <NUM> for which the measurement <NUM> is performed. In some embodiments, the Majorana hexon <NUM> may include at least one coherent superconducting link between a pair of non-adjacent MZMs <NUM> in order to facilitate coherent single-electron tunneling between the non-adjacent MZMs <NUM>.

As shown in the examples of <FIG> and <FIG>, a plurality of Majorana hexons <NUM> may be included in the topological quantum computing device <NUM>. Multi-qubit operations may be performed by weakly coupling MZMs <NUM> from different Majorana hexons <NUM> to common quantum dots <NUM>. Since the coupling is weak, the charging energy protection against quasi-particle poisoning may remain effective during such operations. The charging energy EC of each Majorana hexon <NUM> may be maintained during measurements <NUM> by performing measurements <NUM> of operators that commute with the charging energy EC. These measurements <NUM> may be those that include even numbers of Majorana operators, as discussed in further detail below.

Each of the six MZMs <NUM> included in a Majorana hexon <NUM> may be labeled with subscripts <NUM> through <NUM> and may each be associated with a Majorana fermionic operator γj at the jth position. The operators γj may each obey the fermionic anticommutation relation {γj, γk} = <NUM>δjk. For any ordered pair of MZMs <NUM> j and k, their joint fermionic parity operator is given by iγjγk = -iγkγj, which has eigenvalues pjk = ±<NUM> for even and odd parity respectively. The fermionic parity of an MZM <NUM> is a topological charge (also known as a fusion channel) included in the quantum state <NUM> when the topological quantum computing device <NUM> includes MZMs <NUM>. The corresponding projection operator onto the subspace with parity s = pjk = ±<NUM> is given by equation <NUM> in <FIG>. In addition, the joint fermionic parity operator iγjγk is expressed in terms of the positive and negative projection operators in equation <NUM>. As further shown in <FIG>, the joint fermionic parity operator and the even-parity and odd-parity projection operators can be expressed in a diagrammatic calculus as the diagram elements <NUM>, <NUM>, and <NUM> respectively. The joint fermionic parity operator diagram element <NUM> may be expressed as a wavy line between two straight lines representing the states of MZMs j and k. The joint fermionic parity operator diagram element <NUM> may also be written as an antisymmetric combination of the projectors <NUM> and <NUM>.

In <FIG>, equation <NUM> shows the basis states for a Majorana hexon <NUM>. The ground states of the Majorana hexon <NUM> are assumed to have a fixed value of the collective fermionic parity due to the charging energy of the Majorana hexon <NUM>. In the example of <FIG>, the ground states of the Majorana hexon <NUM> each have an even collective fermionic parity p<NUM>p<NUM>p<NUM> = +<NUM>. Thus, states with odd collective parity are excited states associated with quasiparticle poisoning. In this way, the low-energy state space of the hexon is <NUM>-dimensional, with the basis states shown in equation <NUM>. Thus, the Majorana hexon <NUM> may be viewed as a two-qubit system with a first qubit encoded by p<NUM> and a second qubit encoded in p<NUM> in which the basis states shown in equation <NUM> are |<NUM>,<NUM>〉, |<NUM>,<NUM>〉, |<NUM>,<NUM>〉, and |<NUM>,<NUM>〉 respectively. The joint fermionic parity operators may be expressed in terms of Pauli operators on the two qubits as shown in equation <NUM>, where the Pauli matrices are shown in equation <NUM>.

According to the convention used herein, the third MZM 32C and the fourth MZM 32D of the Majorana hexon 30A are used as ancillary MZMs with joint parity p<NUM> = +<NUM>, and the computational qubit is encoded in p<NUM> = p<NUM>. The basis states of the computational qubit are |<NUM>〉 = |p<NUM> = p<NUM> = +〉 and |<NUM>〉 = |p<NUM> = p<NUM> = -〉. The designation of the ancillary MZMs 32C and 32D and the computational MZMs 32A, 32B, 32E, and 32F is represented diagrammatically in diagram <NUM> of <FIG>. A general logical qubit state may be represented by the diagram <NUM> in <FIG>, in which the logical qubit state is represented as a weighted sum of basis states.

A single-qubit Clifford gate may be implemented on the computational qubit in a topologically protected manner via a "measurement-only" braiding protocol. The braiding transformations may be represented in term of Majorana operators as shown in equation <NUM> of <FIG>. Equation <NUM> shows a counterclockwise exchange of MZMs <NUM> at positions j and k. Using the measurement-only protocol, the single-qubit Clifford gate may be realized by sequentially measuring the joint fermionic parities of MZM pairs. These sequential measurements may be subject to the following constraints. First, the first measurement involves exactly one MZM <NUM> from the ancillary pair. Second, subsequent measurements involve exactly one MZM <NUM> from the preceding measured pair. Third, the final measurement involves the original ancillary pair and the measurement outcome equals the ancillary pair's initial joint parity, which is taken to be p<NUM> = +<NUM> as discussed above. Thus, sequential measurements correspond to anti-commuting parity operators, where measurements of pairs (jk) and (lm) are allowed to follow one another if and only if iγjγkiγlγm = -iγlγmiγjγk. Each measurement sequence may be viewed as a sequence of anyonic teleportations where, in each measurement, the encoded qubit state is re-encoded in a different set of MZMs <NUM>, and where the measured pair of MZMS <NUM> temporarily becomes the ancillary pair. In this view, the sequence of teleportations defines the braiding "path" and enacts the corresponding braiding transformation on the encoded state.

In some embodiments, the topological quantum computing device <NUM> may be configured to implement the logic gate <NUM> at least in part by performing a forced measurement of a projection operator <NUM> included in the logic gate <NUM>. In such embodiments, the forced measurement may include performing a first measurement of a joint fermionic parity operator iγjγk of the quantum state <NUM>, as discussed above. Performing the forced measurement may further include determining whether a result of the first measurement is a predetermined target value <NUM>. For example, the predetermined target value <NUM> may be the initial joint parity of the ancillary pair. When the result of the first measurement is not the predetermined target value <NUM>, performing the forced measurement may further include repeating a second measurement <NUM> that was performed prior to the first measurement <NUM> in the measurement sequence <NUM>. Repeating the second measurement <NUM> may reset the quantum state <NUM> to a state prior to the first measurement <NUM> in the measurement sequence <NUM>. Performing the forced measurement may further include repeating the first measurement <NUM> of the joint fermionic parity operator. Thus, the joint fermionic parity operator iγjγk may be repeatedly measured until it is measured to be equal to the predetermined target value <NUM>.

A sequence of projectors on a Majorana hexon <NUM> subject to the above constraints may generate a single-qubit Clifford gate acting on the encoded computational qubit. <FIG> further shows an example of a measurement sequence <NUM> in equation <NUM>, where S is the π/<NUM> phase gate. The initial measurement of the ancillary pair shown in equation <NUM> is redundant when the ancillary MZMs <NUM> are initialized with the positive parity. As another example, equation <NUM> shows the gate B = S†HS†, where H is the Hadamard gate, acting on the qubit. The gate set {S, B} is a generating set for all single-qubit Clifford gates C<NUM>.

<FIG> shows an example diagram <NUM> of a measurement sequence <NUM> that produces the π/<NUM> phase gate S discussed above. <FIG> further shows an example diagram <NUM> of a measurement sequence <NUM> that produces the B gate. The initial measurements of the ancillary pair are omitted in both diagram <NUM> and diagram <NUM>, since the parity of the ancillary pair is assumed to be initialized to p<NUM> = +<NUM>.

Two-qubit gates may be generated from sequences of <NUM>-MZM and/or <NUM>-MZM projection operators. Turning now to <FIG>, an ancillary projection operator for a two-qubit operation is defined in equation <NUM>. The two-qubit operation may be performed at two Majorana hexons 30A and 30B. Here, the MZMs <NUM>, <NUM>, 32I, 32J, <NUM>, and <NUM> included in the second Majorana hexon 30B are labeled <NUM>' through <NUM>' to distinguish them from the MZMs 32A, 32B, 32C, 32D, 32E, and 32F included in the first Majorana hexon 30A. <FIG> further shows, in equation <NUM>, a definition of the ancillary projection operator for operations on more than two qubits.

In addition, <FIG> shows a definition for a parity operator in equation <NUM>. In equation <NUM>, M is a set of <NUM>N MZMs <NUM> of N Majorana hexons <NUM>, including two MZMs <NUM> from each Majorana hexon <NUM>. Since the particle number on each Majorana hexon <NUM> is preserved, each measurement operator involves an even number of Majorana operators on each Majorana hexon <NUM>. <FIG> further shows a definition of a projection operator on the set M. In equation <NUM>, s = ±<NUM>.

The ancillary projection operator may begin each measurement sequence <NUM> when a two-qubit operation is performed. In addition, the ancillary projection operator may end each measurement sequence <NUM> when the two-qubit operation is performed. In the case of two-qubit operations, each measurement sequence <NUM> may also end with the ancillary projection operator so that both ancillary pairs end up in their respective initialized states. However, if <MAT> or <MAT> commutes with every term in the measurement sequence <NUM>, then the final measurement of the respective ancillary pairs does not need to involve the corresponding measurement pairs of MZMs <NUM>, since the measurement pairs will already be in the final ancillary state.

A system of N Majorana hexons <NUM> may encode N computational qubits, which is a space of dimension <NUM>N. The general condition for a measurement sequence <NUM> of fermionic parity measurements involving N Majorana hexons <NUM> to compile to a unitary gate acting on the computational qubits is that the measurements <NUM> (which may range from <NUM>-MZM to <NUM> N-MZM measurements) do not read information out of the computational state. Thus, the projection operators included in the measurement sequence <NUM> do not collapse the encoded computational state. The measurement sequence <NUM> therefore may not include any projector sequences that multiply out to an operator of rank less than <NUM>N.

The relation between projector sequences and logic gates <NUM> is many-to-one. A specific sequence of projectors, corresponding to a specific sequence of measurements and outcomes (or forced-measurements), used to generate a gate with a measurement-only protocol may be denoted as shown in equation <NUM> of <FIG>. In equation <NUM>, the labels Mµ are used to denote an allowed set of an even number of MZMs <NUM> whose joint fermionic parity is projected onto corresponding parity sµ at the µth projector in the sequence. The resulting unitary gate G acting on the encoded computational state space is shown in equation <NUM> of <FIG>.

In embodiments in which the logic gate <NUM> is constructed from two Majorana hexons 30A and 30B, the Hilbert space of the two Majorana hexons 30A and 30B is the tensor product of that of the two Majorana hexons 30A and 30B. <FIG> shows, in equation <NUM>, the quantum state <NUM> formed by the tensor product of the respective states of the first Majorana hexon 30A and the second Majorana hexon 30B.

In order to generate entangling two-qubit gates, one or more measurements <NUM> may be made of the collective fermionic parity of four MZMs <NUM>. The <NUM>-MZM joint parity projector is shown in equation <NUM> of <FIG>, where the measurement <NUM> is performed on the MZMs <NUM> labeled j and k from the first Majorana hexon 30A and the MZMs <NUM> labeled l' and m' from the second Majorana hexon 30B. Similarly to the <NUM>-MZM joint parity projector shown in equation <NUM>, the <NUM>-MZM joint parity projector of equation <NUM> does not change the total fermionic parity of either Majorana hexon 30A or 30B.

Equation <NUM> of <FIG> shows an example of a two-qubit entangling gate W. A sequence of projectors that may be measured to obtain the gate W is shown in equation <NUM>. Depending on the outcomes of measuring the projection operators shown in equation <NUM>, either W may be obtained (when s<NUM>s<NUM>s<NUM> = +<NUM>) or the inverse of W (when s<NUM>s<NUM>s<NUM> = -<NUM>) may be obtained. The first term in the tensor product acts on the ancillary qubits and the second acts on the computational qubits. Since <MAT> commutes with every operator included in the measurement sequence <NUM> of equation <NUM>, the final projector only acts on the ancillary pair of MZMs <NUM>. Equation <NUM> shows an expansion of the sequence of projection operators shown in equation <NUM>. In equation <NUM>, the ancillary projection operator <MAT> is factored out, since it commutes with each of the other projection operators in the measurement sequence <NUM>.

The gate set {S, B, W}, where the single-qubit gates can act on any qubit and the two-qubit gates can act on any nearest-neighbor pair of qubits, may generate any N-qubit Clifford gate CN. For example, the controlled-Z gate may be obtained as C(Z) = (S† ⊗ S†)W, and the controlled not gate may be obtained from C(Z) by conjugating the second qubit by H = SBS. Thus, since {S, B, C(Z)} generates the entire set of N-qubit Clifford gates for any N, the gate set {S, B, W} also does.

As discussed above, some measurements <NUM> may be more difficult to perform than other measurements <NUM>. These differences in measurement difficulty may include differences in error rates of the measurements <NUM>. Additionally or alternatively, other factors may be used to determine the difficulty of a measurement <NUM>. The error rate of a measurement <NUM> may be affected by a distance between the MZMs <NUM> on which the measurement <NUM> is performed. In some embodiments, the difficulty of a measurement <NUM> may be lower when the MZMs <NUM> being measured are closer to each other in the lattice. The processor <NUM> may be configured to assign respective estimated weighted resource costs <NUM> to the measurements <NUM> to account for these differences in difficulty.

In the two-sided Majorana hexon architecture <NUM> of <FIG> and the one-sided Majorana hexon architecture <NUM> of <FIG>, measurements <NUM> are performed by coupling MZMs <NUM> to quantum dots <NUM>. <FIG> show three example fermionic parity measurement configurations <NUM>, <NUM>, and <NUM>. In the examples of <FIG>, the couplings between the Majorana hexons <NUM> and the quantum dots <NUM> form interference loops delineated by the paths connecting the MZMs <NUM> through the Majorana hexon <NUM> and the paths connecting MZMs <NUM> through the quantum dots <NUM>. To select the interference paths, electrostatic depletion gates are provided such that different parts of the semiconductor may be connected or disconnected. These gates are referred to as cutter gates <NUM>.

Each cutter gate <NUM> that is opened to form the semiconductor quantum dot configuration may increase the difficulty, as the number of open cutter gates <NUM> used to realize a quantum dot <NUM> affects the size of the quantum dot <NUM>. When the number of open cutter gates <NUM> increases, the coherence of the quantum dot <NUM> may decrease, thus adding a source of noise to the measurement. In addition, the overall length of the semiconducting path may affect phase coherence, and the volume of the semiconducting region enclosed by the path may affects properties of the quantum dot <NUM> such as its charging energy and level spacing. Measurements are typically easier to perform for smaller quantum dots <NUM>.

The plurality of cutter gates <NUM> included in a semiconducting path may include one or more vertical cutter gates 76A. The plurality of cutter gates <NUM> may further include one or more horizontal cutter gates 76B. In the fermionic parity measurement configuration <NUM> of <FIG>, the semiconducting path includes two vertical cutter gates 76A; in the fermionic parity measurement configuration <NUM> of <FIG>, the semiconducting path includes seven vertical cutter gates 76A; and in the fermionic parity measurement configuration <NUM> of <FIG>, the semiconducting path includes one vertical cutter gate 76A. The length of the semiconducting path may, in some embodiments, be proportional to the number of vertical cutter gates 76A included in the semiconducting path.

Wherever an MZM <NUM> couples to the semiconductor <NUM>, the coupling may be tuned by a cutter gate <NUM> forming a tunnel junction <NUM>. In contrast to cutter gates <NUM> between semiconducting regions, which are typically either fully opened or closed, each tunnel junction <NUM> may be tuned such that the ratio of the coupling energy of the tunnel junction <NUM> to the charging energy EC of the MZM <NUM> is in a regime in which the effect of the MZM state on the quantum dot <NUM> is quickly and reliably measurable, while not suppressing the charging energy of the quantum dot <NUM> or increasing the probability of quasiparticle poisoning. Typically, the visibility of the signal is reduced with each additional tunnel junction <NUM>. Each tunnel junction <NUM> between an MZM <NUM> and a quantum dot <NUM> involved in a measurement <NUM> has an associated tunneling amplitude. An decrease in the tunneling amplitude may decrease the visibility of the measurement, since decreasing the tunneling amplitude may make the energy splitting between states smaller. This decrease in the visibility of the measurement <NUM> may increase the measurement time and/or decrease the precision of the measurement <NUM>.

In addition, noise in the tunneling junction <NUM> may interfere with the measurement signal. As part of the measurement protocol, the coupling between the MZM <NUM> and the semiconductor <NUM> may be tuned from zero to its target value on a timescale that is shorter than the timescale of the measurements <NUM> but slower than a timescale at which diabatic corrections would occur.

The number of tunnel junctions <NUM> involved in a measurement <NUM> may be equal to the number of MZMs <NUM> involved in the measurement <NUM>, since a tunnel junction <NUM> may be provided for each MZM <NUM>. The fermionic parity measurement configuration <NUM> of <FIG> includes two tunnel junctions <NUM>, the fermionic parity measurement configuration <NUM> of <FIG> includes four tunnel junctions <NUM>, and the fermionic parity measurement configuration <NUM> of <FIG> also includes four tunnel junctions <NUM>.

Fluctuations of the background magnetic field may be another source of noise for the measurement. The contribution of these fluctuations to the noise may be proportional to the area enclosed by the interferometric loop delineated by the architecture of the topological quantum computing device <NUM> and the geometry of a given measurement <NUM>. In some embodiments, the hexon architecture geometries may be such that the relevant areas for such errors may be approximately partitioned into integer multiples of a unit area <NUM>. In the fermionic parity measurement configuration <NUM> of <FIG>, the semiconducting path has an area of two times the unit area <NUM>; in the fermionic parity measurement configuration <NUM> of <FIG>, the semiconducting path has an area of seven times the unit area <NUM>; and in the fermionic parity measurement configuration <NUM> of <FIG>, the semiconducting path has an area of three times the unit area <NUM>.

The difficulty of a measurement <NUM> may also depend on the number of Majorana hexons <NUM> involved in the measurement <NUM>. This is because the measurement visibility may be affected by how precisely the quantum state <NUM> may be tuned to the degenerate tunneling point. In addition, the operations utilized in a measurement <NUM> may cause errors that transfer MZMs <NUM> between the different Majorana hexons <NUM>. Increasing the number of Majorana hexons <NUM> involved in a measurement <NUM> may increase the probability of such errors.

Given the factors described above, an estimated weighted resource cost <NUM> of a 2N-MZM measurement <NUM> involving N Majorana hexons <NUM> is shown in <FIG> in equation <NUM>. In equation <NUM>, nc is the number of vertical cutter gates <NUM> opened for the measurement, na is an area enclosed by the interferometry loop delineated by the measurement <NUM> (expressed as an integer multiple of a unit area), and nt is the number of tunneling junctions involved in the measurement <NUM>. The quantities wc, wa, and wt are weighting factors respectively associated with nc, na, and nt. The weights wt associated with the tunneling junctions may also include contributions from the horizontal cutter gates 76B, since the horizontal cutter gates 76B may be used to control tunneling. The effect of the number N of Majorana hexons <NUM> on the estimated weighted resource cost <NUM> is denoted as f(N). Each of wc, wa, wt, and f(N) may be determined experimentally for the particular topological quantum computing device <NUM> at which the measurements <NUM> are performed.

<FIG> show example fermionic parity measurement configurations <NUM>, <NUM>, <NUM>, <NUM>, and <NUM> that may be used with the one-sided Majorana hexon architecture <NUM> shown in <FIG>. The fermionic parity measurement configuration <NUM> of <FIG> has three vertical cutter gates 76A, two tunneling junctions <NUM>, and an interference loop enclosing an area of three times the unit area <NUM>. The fermionic parity measurement configuration <NUM> of <FIG> has three vertical cutter gates 76A, four tunneling junctions <NUM>, and an interference loop enclosing an area of five times the unit area <NUM>. The fermionic parity measurement configuration <NUM> of <FIG> has zero vertical cutter gates 76A, four tunneling junctions <NUM>, and an interference loop enclosing an area of two times the unit area <NUM>. The fermionic parity measurement configuration <NUM> of <FIG> has two vertical cutter gates 76A, eight tunneling junctions <NUM>, and an interference loop enclosing an area of six times the unit area <NUM>. The fermionic parity measurement configuration <NUM> of <FIG> has four vertical cutter gates 76A, eight tunneling junctions <NUM>, and an interference loop enclosing an area of four times the unit area <NUM>. The example fermionic parity measurement configurations <NUM>, <NUM>, <NUM>, and <NUM> of <FIG> respectively are configurations of corresponding measurements <NUM> in an upward direction, a downward direction, a rightward direction, and a leftward direction.

So far, the MZMs <NUM> in a Majorana hexon <NUM> have been labeled <NUM>,. ,<NUM> and have been assigned roles in the measurement <NUM> according to these labels. For example, in the computational basis, the MZMs <NUM> labeled as <NUM> and <NUM> serve as the ancillary pair, while MZMs <NUM>, <NUM>, <NUM>, and <NUM> collectively encode the computational qubit. However, the six labels may be assigned to the physical MZMs <NUM> of a Majorana hexon <NUM> according to other labeling schemes. The choice of labeling scheme may affect the difficulty of a measurement <NUM>, as discussed below. In some embodiments, the processor <NUM> may be configured to determine the estimated total resource cost <NUM> of at least one measurement sequence <NUM> at least in part by relabeling the MZMs <NUM> included in the Majorana hexon <NUM> to change which MZMs <NUM> are included in the computational qubit and which MZMs <NUM> are included in the ancillary qubit. In embodiments in which the topological quantum computing device <NUM> includes one or more Majorana tetrons <NUM> or Majorana octons, the MZMs <NUM> included in the Majorana tetrons <NUM> or Majorana octons may also be relabeled change which MZMs <NUM> are included in the computational qubit and which MZMs <NUM> are included in the ancillary qubit.

In the following example, let 〈a, b, c, d, e, f〉 denote the configuration of MZMs <NUM> within a Majorana hexon <NUM>, where for one-sided Majorana hexons, the labeling goes from top to bottom as shown in <FIG>, and for two-sided Majorana hexons, the labeling goes counterclockwise from the top-left to the top-right as shown in <FIG>. A possible configuration for either Majorana hexon architecture, as discussed above, is (<NUM>, <NUM>, <NUM>, <NUM>, <NUM>, <NUM>). Here, MZMs <NUM> and <NUM> are on opposite ends of the Majorana hexon <NUM>. On the other hand, in the configuration 〈<NUM>, <NUM>, <NUM>, <NUM>, <NUM>, <NUM>〉, the MZMs <NUM> and <NUM> are adjacent. Thus, different configurations of MZMs <NUM> will result in different assignments of estimated weighted resource costs <NUM> to measurements <NUM>. For example, a measurement of MZMs <NUM> and <NUM> may have estimated weighted resource costs <NUM> of w(<NUM>)〈<NUM>,<NUM>,<NUM>,<NUM>,<NUM>,<NUM>〉 and w(<NUM>)〈<NUM>,<NUM>,<NUM>,<NUM>,<NUM>,<NUM>〉, the latter of which may be higher. If this measurement <NUM> occurs frequently in a measurement sequence <NUM>, it may be less resource-intensive to use the configuration (<NUM>, <NUM>, <NUM>, <NUM>, <NUM>, <NUM>). In addition, measurements <NUM> involving pairs that are neighbors in different directions may have different estimated weighted resource costs <NUM>. For example, in a one-sided Majorana hexon architecture <NUM>, measurements <NUM> connecting vertical neighbors may be less resource-intensive than measurements <NUM> connecting horizontal neighbors.

The two-sided Majorana hexon architecture <NUM> and the one-sided Majorana hexon architecture <NUM> each have symmetry relations that may reduce the number of labeling configurations to evaluate. A two-sided Majorana hexon has horizontal and vertical reflection symmetry, which reduces the number of inequivalent configurations from <NUM>! = <NUM> to <NUM>. One-sided Majorana hexons have vertical reflection symmetry, which reduces the number of inequivalent configurations from <NUM> to <NUM>.

In order for implementation of the logic gate <NUM> to be scalable, the full array of Majorana hexons <NUM> in the topological quantum computing device <NUM> may utilize labeling configurations that are periodic in the array. In some embodiments, each Majorana hexon <NUM> in the array may use the same labeling configuration. However, different configurations may be assigned to different Majorana hexons <NUM> in other embodiments. For example, the array may include a first configuration for all right-facing one-sided Majorana hexons <NUM> and a second configuration for all left-facing one-sided Majorana hexons <NUM>.

In some embodiments, the topological quantum computing device <NUM> may be configured to implement the logic gate <NUM> at least in part by performing a forced measurement of a projection operator <NUM> included in the logic gate <NUM>. The forced measurement may include performing a first measurement <NUM> of a joint fermionic parity operator ΓM of the quantum state <NUM>. When the joint fermionic parity operator ΓM of an ordered set of M MZMs <NUM> is measured in a system in a pure state |Ψ〉, the measurement outcome s = ± may be obtained with the probability shown in equation <NUM> in <FIG>. A post-measurement state shown in equation <NUM> is obtained when this measurement <NUM> is made. For general states described by a density matrix ρ, the measurement outcome s may be obtained with the probability shown in equation <NUM>. The post-measurement state for general states described by the density matrix ρ is shown in equation <NUM>.

The forced measurement may further include determining whether a result of the first measurement <NUM> is a predetermined target value <NUM>. Since the predetermined target value <NUM> of s is not always obtained when the first measurement <NUM> is performed, the projection operator <NUM> with the target parity is not always obtained. In order to obtain the target projection operator <MAT> in the measurement-only scheme, a repeat-until-success "forced measurement" procedure may be used. When a measurement of iγjγk, the joint fermionic parity operator of the target projection operator <MAT>, is performed, the probability of obtaining the target parity may be <NUM>/<NUM>. The initial measurement <NUM> of the ancillary pair of MZMs <NUM> may have a deterministic outcome. When the result of the first measurement <NUM> of iγjγk is not the predetermined target value <NUM>, the forced measurement may further include resetting the quantum state <NUM> to a state prior to the first measurement <NUM>. The quantum state <NUM> may be reset by performing a parity measurement on the pair of MZMs measured in a second measurement <NUM> that was performed prior to the first measurement <NUM>.

Subsequently to resetting the quantum state <NUM>, performing the forced measurement may further include repeating the first measurement <NUM> of the joint fermionic parity operator iγjγk. If the repetition of the first measurement <NUM> also does not return the predetermined target value <NUM> of the parity, the steps above may be repeated. Thus, the joint fermionic parity operator iγjγk may be measured and reset until the measurement <NUM> returns the predetermined target value <NUM>.

<FIG> shows a flowchart of an example method <NUM> by which the forced measurement may be performed. At step <NUM>, the method <NUM> may include performing a measurement <NUM> of a joint fermionic parity operator of the quantum state. At step <NUM>, the method <NUM> may further include determining whether a result of the measurement <NUM> is a predetermined target value <NUM> of the parity. If the result of the measurement is not the predetermined target value <NUM>, the method <NUM> may include, at step <NUM>, resetting the quantum state <NUM>. The method <NUM> may then return to step <NUM>. If the result of the measurement <NUM> is the predetermined target value <NUM>, the method <NUM> may further include, at step <NUM>, using a projection operator with the predetermined target value <NUM> in the measurement sequence <NUM>.

An example in which forced measurement is used to obtain a measurement sequence <NUM> is shown in <FIG>. In the example of <FIG>, the S gate is generated with the projection operator sequence <MAT>. A diagram <NUM> for this projector sequence is shown in <FIG>. In the example of <FIG>, the measurement of iγ<NUM>γ<NUM> results in the target value <NUM> of s<NUM> = +, but the measurement of iγ<NUM>γ<NUM> results in the undesired value of s<NUM> = -, as shown in diagram <NUM> in <FIG>. As shown in diagram <NUM> of <FIG>, the measurement of iγ<NUM>γ<NUM> may be repeated. The quantum state <NUM> is reset regardless of the outcome of this repeated measurement. The measurement of iγ<NUM>γ<NUM> may then be repeated, with another <NUM>/<NUM> probability of obtaining the predetermined target result s<NUM> = -. If the undesired measurement outcome is obtained again, the above steps may be repeated until the predetermined target outcome is obtained. The measurement sequence <NUM> depicted in diagram <NUM> differs only by an overall phase from the measurement sequence <NUM> depicted in diagram <NUM>, as indicated in equation <NUM>.

In order to distinguish the application of a forced-measurement operation from the projectors associated with a physical measurement, the application of this forced-measurement to the MZM pair (jk) in a sequence following a measurement of (kl) is denoted in equation <NUM> of <FIG>. In equation <NUM>, the desired measurement outcome s is obtained at the nth attempt. In addition, sa ≠ s for a = <NUM>,. , n - <NUM>, and the measurement outcomes ra are irrelevant.

The estimated total resource cost <NUM> of the forced measurement sequence of equation <NUM> may be given by equation <NUM> in some embodiments. The estimated total resource cost <NUM> provided in equation <NUM> is equal to the estimated total resource cost <NUM> of an average-case measurement sequence <NUM> including (n) = <NUM> attempts.

As an alternative to the forced measurement procedure discussed above, the following procedure may be used to obtain the predetermined target value <NUM> of a measurement <NUM>. When the measurement <NUM> of the MZM pair (jk) following a measurement <NUM> of the MZM pair (kl) yields an undesired outcome, instead of resetting the quantum state <NUM> by repeating the previous measurement <NUM> of (kl), the quantum state <NUM> may instead be reset by measuring the MZM pair (jl). Thus, resetting the quantum state <NUM> may include, in the alternative forced measurement protocol, measuring an MZM pair including a first MZM <NUM> (labeled above as j) on which the first measurement <NUM> was performed. The MZM pair further includes a second MZM <NUM> (labeled above as l) on which a second measurement <NUM> that was performed prior to the first measurement <NUM> in the measurement sequence <NUM> was performed. However, unlike in the first forced measurement protocol discussed above, the second measurement <NUM> was not performed on the first MZM <NUM> in the MZM pair.

In more general terms not specific to topological quantum computing devices <NUM> that include MZMs <NUM>, resetting the quantum state <NUM> according to the alternative forced measurement protocol includes measuring a plurality of topological charges. The plurality of topological charges including a symmetric difference of a first plurality of topological charges on which the first measurement <NUM> was performed and a second plurality of topological charges on which a second measurement <NUM> was performed prior to the first measurement <NUM> in the measurement sequence <NUM>. "Symmetric difference" is defined as the union of two sets minus the intersection of the two sets.

The alternative forced measurement approach is shown diagrammatically in <FIG> for the projector sequence <MAT>, when an undesired measurement outcome occurs for the measurement of MZMs (<NUM>). <FIG> shows, at diagram <NUM>, the projector sequence after the undesired measurement outcome s<NUM> = - has occurred. <FIG> further shows a first forced measurement sequence in diagram <NUM>. <FIG> shows, in diagram <NUM>, a second forced measurement sequence that may be used as an alternative to that of <FIG>. In addition, as shown in equation <NUM> in <FIG>, the projector sequence shown in diagram <NUM> differs only by an overall constant from the projector sequence <MAT>.

In order to differentiate the alternative forced-measurement protocol from the first forced measurement protocol (and from an ordinary projector), this alternative forced measurement protocol as applied to the MZM pair (jk) following a measurement <NUM> of the MZM pair (kl) is defined in equation <NUM> of <FIG>. In equation <NUM>, sa ≠ s for a = <NUM>,. , n - <NUM>, and the measurement outcomes pa are irrelevant. As in the first forced measurement protocol discussed above, the estimated total resource cost <NUM> of the forced measurement sequence of equation <NUM> may be the geometric mean of the weighted estimated resource costs <NUM> of the measurements <NUM> in the measurement sequence <NUM>. This estimated total resource cost <NUM> may be equal to the difficulty weight of the average-case measurement sequence <NUM> including (n) = <NUM> attempts. The estimated total resource cost <NUM> in such embodiments is given in equation <NUM> of <FIG>. This alternative forced measurement protocol may be less resource intensive than the first forced measurement protocol in embodiments where parity measurements of MZMs (jl) have lower weighted estimated resource costs <NUM> than measurements of MZMs (kl).

Forced measurement protocols for 2N-MZM measurements, in particular <NUM>-MZM measurements, are discussed below with reference to <FIG>. These forced measurement protocols may also be used for <NUM>-MZM measurements that follow a <NUM>-MZM measurement. Equation <NUM> of <FIG> shows the condition under which a forced measurement of M<NUM> follows a measurement of M<NUM> for some choice of M<NUM>. As another condition for performing the forced measurement on M<NUM>, the subsequent projection operators in the measurement sequence <NUM> may not commute with each other. This condition holds when ΓM<NUM>ΓM<NUM> = -ΓM<NUM>ΓM<NUM> and ΓM<NUM>ΓM<NUM> = -ΓM<NUM>ΓM<NUM>.

Another form of the left-hand side of equation <NUM> is shown in equation <NUM> of <FIG>. Equation <NUM> further shows the conditions under which equation <NUM> holds. Under the conditions given in equation <NUM>, ΓM<NUM> may be replaced with a constant. <FIG> provides a generalization of the two different forced measurement protocols described above. The second condition shown in equation <NUM> may lead to invalid measurement sequences that would collapse the qubit states or to measurements of greater than <NUM>N-MZMs if M<NUM> and M<NUM> include more than two elements each. Since the estimated total resource cost <NUM> grows quickly as the number of MZMs <NUM> increases, it is preferable to avoid having to measure such sequences. However, the above problems do not arise when the first condition of equation <NUM> is satisfied.

Equation <NUM> of <FIG> shows four example forced measurement protocols for measurement sequences <NUM> involving four MZMs <NUM>. In addition, equation <NUM> shows the respective estimated total resource costs <NUM> of the measurement sequences <NUM> of equation <NUM>, according to one embodiment.

The forced measurement protocols discussed above provides control over which fermionic parities are projected upon at each measurement <NUM> in the measurement sequence <NUM>. These forced measurement protocols allow for the implementation of a projector sequence that generates a specified target logic gate <NUM>. A forced measurement protocol could be applied for every projector in a given projector sequence. However, such a strategy may be inefficient, since the different projectors in the measurement sequence <NUM> may have a correlated effect on the resulting logic gate <NUM>. When determining the estimated total resource cost <NUM> for a measurement sequence, the processor <NUM> may be further configured to determine which projectors in a measurement sequence <NUM> have a correlated effect and, therefore, which specific measurements <NUM> can tolerate any outcome and which measurements <NUM> may have to be forced in order to obtain the logic gate <NUM>. In such embodiments, the processor <NUM> may be configured to determine the respective estimated total resource cost <NUM> of each measurement sequence <NUM> at least in part by identifying one or more measurement sequences <NUM> that differ by an overall Pauli operator, as discussed in further detail below.

Returning to <FIG>, equation <NUM> shows the joint fermionic parity operators iγjγk expressed in terms of the Pauli operators shown in equation <NUM>. In addition, as shown in <FIG>, a projection operator <MAT> may be drawn as a cap and cup, connected by a wavy line if s = - and unconnected if s = +. For every s = - projector in the measurement sequence, the corresponding fermion line (that terminates on two MZM lines) may be moved to the top of the diagram using the diagrammatic rules. Each such fermion line that has been slid to the top of the diagram simply connects two MZM lines j and k, resulting in a joint fermionic parity operator iγjγk. In embodiments in which every measurement sequence starts and ends with a forced measurement of <MAT>, the fermion lines do not connect to the ancillary MZM lines when pushed to the top of the diagram. Thus, in such embodiments, j and k do not correspond to ancillary MZMs <NUM>.

Turning now to <FIG>, the fermion lines that are slid to the top of the diagram correspond to the Pauli operators shown in table <NUM>. The complete operation effected on the computational subspace by a measurement sequence <NUM> may be a braiding transformation (hence a Clifford gate) determined by which MZMs <NUM> are measured in the measurement sequence <NUM>, followed by a Pauli gate determined by the measurement outcomes. Equation <NUM> of <FIG> shows a single-hexon projector sequence that compiles to a Clifford gate G. In equation <NUM>, the projection channel parities sµ need not all be positive. Equation <NUM> shows the single-hexon projector sequence of equation <NUM> when respective parities of all the projection operators have been set to positive. Equation <NUM> may be rewritten as equation <NUM>, where q is the number of projection operators with negative parities in equation <NUM>, p is an integer, and P is a Pauli gate. Thus, the effect of the measurement outcomes sµ in a single-hexon projector sequence is to change the resulting logic gate <NUM> by at most a Pauli gate.

<FIG> shows, in equation <NUM>, a projection operator sequence that may be used to implement any of the Pauli gates. In equation <NUM>, the resulting Pauli gate P is independent of s<NUM> and s<NUM>. This independence is shown in <FIG> with the diagram <NUM>, which differs only by overall phases from the diagrams <NUM> and <NUM> of <FIG>. Isotopy of the MZM lines allows them to be straightened, leaving no nontrivial braiding in diagram <NUM>. Thus, P+ = <NUM>, where P+ is the Pauli gate that results when all measurement outcomes sµ = + in equation <NUM>. In addition, in <FIG>, both ends of the s<NUM> line connect to the j = <NUM> MZM line when straightened and both ends of the s<NUM> line connect to the j = <NUM> MZM line when straightened, which allows the s<NUM> and s<NUM> lines to be removed without affecting the resulting logic gate <NUM>. After straightening the MZM lines and sliding the sµ lines to the top of the diagram, s<NUM> = - contributes the operator iγ<NUM>γ<NUM> = Z ⊗ X, s<NUM> = - contributes the operator iγ<NUM>γ<NUM> = <NUM> ⊗ X, and s<NUM> = - contributes the operator iγ<NUM>γ<NUM> = <NUM> ⊗ Z. Thus, the compiled Pauli gate is P as shown in equation <NUM>.

For multi-hexon projector sequences, changing the projection channel parities sµ also changes the resulting logic gate <NUM> by at most a multi-qubit Pauli gate. In addition, by tracking the effects of the projection channel parities sµ on the resulting compiled gate, the processor <NUM> may be configured to determine one or more measurements <NUM> in the measurement sequence <NUM> for which to perform forced measurements. For a single-hexon projector sequence, when all the fermion lines in the projector sequence are moved to the top of the diagram, each line may either be removed or may end up in one of the six configurations connecting MZM lines represented by the fermion parity operators iγjγk listed in table <NUM>. In this way, the specific Pauli operator contributed by a given measurement outcome to P in the decomposition shown in equation <NUM> may be determined. Therefore, any Clifford gate may be generated from a measurement sequence <NUM> in which three or fewer of the measurements <NUM> in the measurement sequence <NUM> are forced measurements. Of these three or fewer forced measurements, one forced measurement may be used to assign a positive parity to the ancillary pair of MZMs <NUM>, and the other two or fewer forced measurements may be used to obtain a target Pauli gate. For example, the measurement sequence <NUM> of <FIG> may generate a particular target Pauli gate for any values of s<NUM>, s<NUM>, and s<NUM> by choosing s<NUM> and s<NUM> appropriately via forced measurements.

Another approach that may be used in addition to, or as an alternative to, forced measurement in measurement-only topological quantum computing is described below. In this approach, known as Majorana-Pauli tracking, measurement outcomes that only change the resulting braiding transformations by Pauli gates may be tracked. More generally, a similar tracking strategy may be employed when the measurement outcomes are Abelian anyons. Majorana-Pauli tracking may allow for the use of fewer physical measurement operations and may allow deterministic measurement sequences <NUM> to be used for topological gate operations.

<FIG> shows the construction of a measurement sequence <NUM> that compiles to a gate G acting on the computational state space in a system of N Majorana hexons <NUM>. First, in equation <NUM>, an ancillary projection operator <NUM> is defined for the N-hexon system. In addition, equation <NUM> introduces a reversal operator that flips the state of each ancillary qubit whose initial and final projections differ. In equation <NUM>, γa,j is the ath MZM <NUM> of the jth Majorana hexon <NUM>. In equation <NUM>, the ancillary projection operator of equation <NUM> is expressed in terms of the reversal operator of equation <NUM>. Equation <NUM> provides the measurement sequence <NUM> by which the gate G is compiled according to the Majorana-Pauli tracking protocol, in terms of the operators defined in equations <NUM>, <NUM>, and <NUM>. The operator G shown in equation <NUM> is a unitary operator and therefore does not reduce the rank of the computational subspace.

When Majorana-Pauli tracking is used, instead of the convention p<NUM> = +<NUM> used with the forced measurement protocol, p<NUM> may take either a positive or negative value. The value of p<NUM> may change over the course of generating a measurement-only gate. As in the forced measurement protocol, the computational qubit is encoded in p<NUM>. Thus, when the collective fermionic parity of the Majorana hexon <NUM> is even, the remaining parity p<NUM> is determined by the respective parities of the other two pairs according to p<NUM> = p<NUM>p<NUM>. For the general case in which the Majorana hexon <NUM> may have a collective fermionic parity ph that may be even or odd, the parity p<NUM> is given by p<NUM> = p<NUM>p<NUM>ph.

As shown in equation <NUM>, an N-qubit Pauli operator may be applied to the gate G of equation <NUM> to produce the same gate with a different sequence of projections. Thus, if a measurement-only sequence of measurements <NUM> is performed to obtain a logic gate <NUM> and the measurement outcomes are tracked, the resulting logic gate <NUM> will have a known Pauli gate correction. If the non-Clifford gates that are utilized in a quantum computation are single-qubit phase gates (in any of the Pauli bases), the Pauli gate correction may be pushed through the phase gates with at most a Clifford gate correction. When a Clifford gate correction is performed, a Clifford gate by which the logic gate <NUM> differs from the target logic gate may be tracked as the measurement sequence <NUM> is performed. In such embodiments, the Clifford gate correction may be dealt with by updating the subsequent Clifford gate in the computation. When non-Clifford phase gates are implemented by injecting states, such a Clifford correction would typically be used. Thus, the effect of the Clifford correction on performance would be small.

Approaches are discussed below by which the processor <NUM> may be further configured to determine a first measurement sequence <NUM> that has a lowest estimated total resource cost <NUM> of the plurality of measurement sequences <NUM>. The lowest estimated total resource cost <NUM> may be a global minimum estimated total resource cost <NUM> across all measurement sequences <NUM> that implement the logic gate <NUM> or may alternatively be a minimum estimated total resource cost of a subset of all such measurement sequences <NUM> that are searched by the processor <NUM>.

When a topological quantum computation is performed using a plurality of different logic gates <NUM>, the processor <NUM> may be configured to select the lowest estimated total resource cost <NUM> for a subset of the plurality of different logic gates <NUM>. Within a given topological quantum computation, it is not always possible for all logic gates <NUM> involved in the computation to be implemented with their respective first measurement sequences <NUM>. In some embodiments, the first measurement sequence <NUM> may be used for one or more logic gates <NUM> that are used with a high frequency in the topological quantum computation. For example, the one or more logic gates <NUM> for which the first measurement sequence <NUM> is determined may be the controlled-Pauli gates, the Hadamard gate, or all single-qubit Clifford gates.

When the processor <NUM> searches for the first measurement sequence <NUM>, the processor <NUM> may perform the search over measurement sequences <NUM> used in the Majorana-Pauli tracking protocol or the forced measurement protocol. Determination of the first measurement sequence <NUM> having the lowest estimated total resource cost <NUM> is first discussed herein for the Majorana-Pauli tracking protocol. When a measurement sequence <NUM> is compiled in terms of the projector sequence shown in equation <NUM>, the sequence of physical measurements that will be performed is the sequence M<NUM>,. , Mn specified in the projector sequence. When the physical measurement outcomes do not match the specified projector channels sµ, the resulting gate differs from G by at most a Pauli gate, which may be tracked and compensated for at a later time. As such, this measurement-only realization of G may be assigned the estimated total resource cost <NUM> shown in equation <NUM> of <FIG>.

When Majorana-Pauli tracking is utilized, Clifford gates may be grouped into Pauli cosets given by collections of Clifford gates that are equivalent up to multiplication by an overall multi-qubit Pauli gate. The Pauli coset of an N-qubit Clifford gate G is defined in equation <NUM> of <FIG>. When using Majorana-Pauli tracking, it is not necessary to generate every Clifford gate. Instead, one Clifford gate from each Pauli coset may be generated, since differences by Pauli gates are dealt with by the tracking protocol. Thus, in some embodiments, the most easily realized Clifford gate in a given Pauli coset may be used to implement the entire class of Clifford gates. Thus, the element of each Pauli coset with the lowest estimated total resource cost <NUM> may be used when any Clifford gate in that Pauli coset is called in a computation.

When a forced measurement protocol is utilized instead, the measurement sequence <NUM> for compiling a gate G may be written as shown in equation <NUM> of <FIG>. When determining the first measurement sequence <NUM> with the lowest estimated total resource cost <NUM>, the processor <NUM> may be configured to determine the minimum number of projectors for which forced measurement is needed to generate the gate G with the forced measurement protocol. The projector sequence may then be converted into a measurement sequence <NUM> by utilizing forced measurement for each projector where forced measurement is needed. Standard measurements may be performed for the other projectors. For each projector for which forced measurement is performed, the forced measurement protocol that is used, of the two forced measurement protocols discussed above, may be the forced measurement protocol for which the estimated weighted resource cost <NUM> of the forced measurement is lower.

The estimated total resource cost <NUM> of a measurement sequence <NUM> when a forced measurement protocol is used may be determined by taking the geometric mean of the possible total resource costs of the measurement sequence <NUM>. As shown in equation <NUM> of <FIG>, this geometric mean may be obtained by starting with the expression from equation <NUM> for the difficulty weight of the projector sequence and replacing the weights of the projectors that are forced with the average difficulty weight corresponding to the forced-measurement protocol used. In equation <NUM>, F<NUM> is the set of projectors in the sequence to be implemented by forced measurements of the first type and F<NUM> is the set of projectors in the sequence to be implemented by forced measurements of the second type.

When searching for measurement sequences <NUM> that may be used to implement a logic gate <NUM>, the processor <NUM> may be configured to determine what measurement sequences <NUM> do not collapse the computational state. For single-qubit gates, such measurement sequences <NUM> may satisfy the condition that consecutive <NUM>-MZM measurements must have exactly one MZM <NUM> in common. Under such a condition, each measurement step may involve choosing one MZM <NUM> from the previous measurement pair and one from the four remaining MZMs <NUM>, leading to eight possible measurements <NUM> to choose from. The nth measurement <NUM> in the measurement sequence <NUM> may be constrained to be a measurement of the ancillary pair (<NUM>, <NUM>) of MZMs <NUM>. In addition, the second-to-last measurement <NUM> may be constrained to involve one MZM <NUM> of the preceding pair and one MZM <NUM> of the ancillary pair. Thus, there may be four choices available for the second-to-last measurement <NUM>. The size of the search space for single-hexon measurement sequences of length n may be <NUM><NUM>n-<NUM>. Even though this scaling is exponential in n, the value of n in the first measurement sequence <NUM> with the lowest estimated total resource cost <NUM> is typically low for single-qubit gates.

Once the one or more measurement sequences <NUM> that produce a target logic gate <NUM> without collapsing the computational state are determined, the resulting logic gates G may be evaluated for all possible measurement outcomes sµ. In some embodiments, the processor <NUM> may be configured to perform a brute-force search by determining a respective estimated total resource cost <NUM> for each measurement sequence <NUM> that implements a target logic gate <NUM> and is shorter than a predetermined length <NUM>. For example, the predetermined length <NUM> may be n = <NUM>.

The example provided below discusses determination of the first measurement sequence <NUM> and the lowest estimated total resource cost <NUM> for the set of controlled Pauli gates {C(X), C(Y), C(Z)} and the SWAP gate, which are examples of two-qubit Clifford gates. A quantum state <NUM> of two Majorana hexons <NUM> has <NUM> different nontrivial fermionic parity projectors. In some embodiments, each of the nontrivial fermionic parity projectors may be tested at each measurement step to determine which projectors do not collapse the computational state. In such embodiments, the processor <NUM> may be configured to diagonalize the projector sequence after each time a projector is added. When the projector sequence has been diagonalized, the processor <NUM> may be further configured to discard the projector sequence if the projector sequence collapses the computational state. However, performing the diagonalization of the projector sequence may be computationally costly.

As an alternative, for small values of n, each of the <NUM> possible parity projectors may be applied at each step in the projector sequence. The final resulting logic gates <NUM> generated by the projector sequences may then be checked. In the example in which the respective first measurement sequences <NUM> are determined for each of the controlled Pauli gates, projector sequences that produce the controlled Pauli gates may include at least four projectors. In this example, a search of each measurement sequence <NUM> up to the predetermined length <NUM> of n = <NUM> may be performed.

Additionally or alternatively, each possible projector sequence, up to some predetermined length <NUM>, that includes one <NUM>-MZM projector may be searched for the controlled Pauli gates. For example, the predetermined length <NUM> may be n = <NUM>.

Each projector sequence that compiles to the SWAP gate includes at least two <NUM>-MZM projectors. A search for the two <NUM>-MZM projectors may be performed for projector sequences up to the predetermined length <NUM> of n = <NUM>. For the one-sided Majorana hexon architecture <NUM>, no projector sequences that compile to the SWAP gate are found for this predetermined length <NUM>. Instead, a projector sequence that compiles to the SWAP gate may be formed from a plurality of controlled not gates.

For a two-qubit measurement sequence <NUM> that compiles to a target logic gate <NUM>, the measurement sequence <NUM> may be evaluated for each possible projector channel sµ. Evaluating each measurement sequence <NUM> for each possible projector channel sµ may provide the Pauli correction gate that may be used in the Majorana-Pauli tracking protocol and may also identify the projectors for which forced measurement is performed in the forced measurement protocol.

Correlations between the remaining measurement outcomes may then be identified. Such correlations may be determined starting with the first projector that does not have fixed projection channel, denoted as sv. The subsets of projector sequences where sv = +<NUM> and where sv = -<NUM> may be considered separately. Within each subset, the processor <NUM> may be configured to check whether any subsequent measurement <NUM> has a fixed outcome. If a subsequent measurement <NUM> has a fixed outcome, that measurement <NUM> may be forced onto a channel that is correlated with sv. If the measurement <NUM> does not have a fixed outcome, the above steps may be recursively applied with the projection channel of that measurement replacing sv.

In one example of compiling a two-qubit gate from a measurement sequence <NUM>, the sequence <MAT> compiles to a controlled not gate when s<NUM> = + and s<NUM> = s<NUM>. Thus, the projector sequences <MAT> and <MAT> yield the same logic gate <NUM>. In this example, forced measurement may be performed for µ = <NUM>, <NUM>, <NUM>.

In one example of identifying the first measurement sequence <NUM>, the example weighting factors wc = <NUM>, wa = <NUM>, wt = <NUM>, and f(N) = (N!)N-<NUM> may be used in equation <NUM>. In this example, when either forced measurement or Majorana-Pauli tracking is used with a two-sided Majorana hexon architecture <NUM>, the MZM labeling configuration (<NUM>, <NUM>, <NUM>, <NUM>, <NUM>, <NUM>) yields the lowest estimated total resource cost <NUM> for each of the single-qubit Hadamard gate, the geometric mean of all single-qubit Clifford gates, the geometric mean of controlled not gates acting in all four directions, and the geometric mean of controlled Pauli gates acting in all four directions. For one-sided Majorana hexon architectures <NUM>, when forced measurement is used, the MZM labeling configuration 〈<NUM>, <NUM>, <NUM>, <NUM>, <NUM>, <NUM>〉 yields the lowest estimated total resource cost <NUM> for the Hadamard gate and the geometric mean of all single-qubit Clifford gates, and the MZM labeling configuration 〈<NUM>, <NUM>, <NUM>, <NUM>, <NUM>, <NUM>〉 yields the lowest estimated total resource cost <NUM> for the geometric mean of controlled not gates acting in all four directions and the geometric mean of controlled Pauli gates acting in all four directions. For one-sided Majorana hexon architectures <NUM>, when Majorana-Pauli tracking is used, the MZM labeling configuration (<NUM>, <NUM>, <NUM>, <NUM>, <NUM>, <NUM>) yields the lowest estimated total resource cost <NUM> for each of the single-qubit Hadamard gate, the geometric mean of all single-qubit Clifford gates, the geometric mean of controlled not gates acting in all four directions, and the geometric mean of controlled Pauli gates acting in all four directions.

<FIG> shows a flowchart of an example method <NUM> for performing a quantum computation by implementing a logic gate. The method <NUM> may be used with the quantum computing system <NUM> of <FIG>, the example quantum computing system <NUM> of <FIG>, or some other quantum computing system. At step <NUM>, the method <NUM> includes identifying a plurality of measurement sequences that implement a logic gate. Each measurement sequence includes a plurality of measurements of a quantum state of a topological quantum computing device. In some embodiments, the topological quantum computing device may include at least one of a Majorana tetron including four MZMs, a Majorana hexon including six MZMs, or a Majorana octon including eight MZMs. The topological quantum computing device may include a plurality of Majorana tetrons, hexons, and/or octons in some embodiments. Additionally or alternatively, the topological quantum computing device may include a Majorana tetron including four MZMs and/or a Majorana octon including eight MZMs. A measurement sequence may include measurements of a plurality of Majorana tetrons, hexons, and/or octons in some embodiments.

In some embodiments, step <NUM> may include, at step <NUM>, identifying one or more measurement sequences that implement the logic gate multiplied by an overall Pauli operator. When two measurement sequences differ by an overall Pauli operator, the Pauli operator may be tracked as the measurements in the measurement sequence are performed when the topological quantum computing device implements the logic gate. A correction for the overall Pauli operator may be performed at the end of the measurement sequence. Step <NUM> may allow computational resources to be saved by reducing the number of measurement sequences that are checked by more computationally intensive methods.

At step <NUM>, the method <NUM> further includes determining a respective estimated total resource cost of each measurement sequence of the plurality of measurement sequences. For each measurement sequence of the plurality of measurement sequences, step <NUM> may include, at step <NUM>, determining an estimated weighted resource cost of each measurement included in the measurement sequence. In some embodiments, the estimated weighted resource cost of each measurement may indicate an error rate of the measurement. In embodiments in which step <NUM> is performed, step <NUM> may further include, at step <NUM>, determining the estimated total resource cost of the measurement sequence based on the plurality of estimated weighted resource costs. For example, the estimated total resource cost may be a product of the estimated weighted resource costs of the measurements. In some embodiments, step <NUM> may include determining a respective estimated total resource cost for each measurement sequence that implements the logic gate and is shorter than a predetermined length.

In embodiments in which the quantum state includes a Majorana tetron, Majorana hexon, or Majorana octon, the MZMs included in the Majorana tetron, Majorana hexon, or Majorana octon may have a labeling order that indicates a plurality of MZMs included in a computational qubit and a plurality of MZMs included in an ancillary qubit. In such embodiments, step <NUM> may further include, at step <NUM>, modifying a topological encoding of a computational qubit and an ancillary qubit. These topological encodings may be modified by relabeling the MZMs included in the Majorana tetron, Majorana hexon, or Majorana octon to change how the computational and ancillary qubits are encoded in the physical MZMs.

At step <NUM>, the method <NUM> further includes determining a first measurement sequence that has a lowest estimated total resource cost of the plurality of measurement sequences. At step <NUM>, the method <NUM> further includes implementing the logic gate at the topological quantum computing device by applying the first measurement sequence to the quantum state. In some embodiments, step <NUM> may include the steps of the example method <NUM> for performing a forced measurement shown in <FIG>. In embodiments in which step <NUM> is performed, step <NUM> may further include, at step <NUM>, tracking a Pauli gate correction to the logic gate as the measurement sequence is performed.

Although the above examples are provided for topological quantum computing devices <NUM> including Majorana hexons <NUM>, the systems and methods discussed above may be used when performing topological quantum computation with other non-Abelian anyons or defects. In such embodiments, measurements of fusion channels other than fermionic parities may be performed. The forced measurement and Majorana-Pauli tracking protocols may be used when the measurement outcomes correspond to fusion channels that are Abelian. In addition to MZMs, examples of structures that may be used in the topological quantum computing device include Ising anyons and Parafendleyons (parafermionic zero modes). Although the topological charge discussed in the examples provided above is a joint fermionic parity, the topological charge may be some other quantity when defects other than MZMs are included in the topological quantum computing device. Forced measurement protocols may also be used when the measurement outcomes correspond to non-Abelian fusion channels. In addition, the resource cost estimation systems and methods described above may be applied to other measurement-based operations, such as the injection of non-Clifford gates.

Computing system <NUM> may embody the quantum computing system <NUM> described above and illustrated in <FIG>. Computing system <NUM> may take the form of one or more personal computers, server computers, tablet computers, home-entertainment computers, network computing devices, gaming devices, mobile computing devices, mobile communication devices (e.g., smart phone), and/or other computing devices, and wearable computing devices such as smart wristwatches and head mounted augmented reality devices.

As non-limiting examples, the communication subsystem may be configured for communication via a wireless telephone network, or a wired or wireless local- or wide-area network, such as aHDMI over Wi-Fi connection.

According to a first aspect of the present disclosure, which does not fall under the scope of the claims, a computing system is provided, including a processor configured to identify a plurality of measurement sequences that implement a logic gate. Each measurement sequence may include a plurality of measurements of a quantum state of a topological quantum computing device. The processor may be further configured to determine a respective estimated total resource cost of each measurement sequence of the plurality of measurement sequences. The processor may be further configured to determine a first measurement sequence that has a lowest estimated total resource cost of the plurality of measurement sequences. The topological quantum computing device may be configured to implement the logic gate by applying the first measurement sequence to the quantum state.

According to this first aspect, for each measurement sequence of the plurality of measurement sequences, the processor may be configured to determine the respective estimated total resource cost at least in part by determining an estimated weighted resource cost of each measurement included in the measurement sequence. The processor may be further configured to determine the estimated total resource cost of the measurement sequence based on the plurality of estimated weighted resource costs.

According to this first aspect, the estimated weighted resource cost of each measurement may indicate an error rate of the measurement.

According to this first aspect, the processor may be configured to determine a respective estimated total resource cost for each measurement sequence that implements the logic gate and is shorter than a predetermined length.

According to this first aspect, the processor may be configured to determine the first measurement sequence at least in part by modifying a topological encoding of a computational qubit and an ancillary qubit.

According to this first aspect, the topological quantum computing device may include a plurality of Majorana zero modes (MZMs).

According to this first aspect, the topological quantum computing device may include at least one of a Majorana tetron including four MZMs, a Majorana hexon including six MZMs, or a Majorana octon including eight MZMs.

According to this first aspect, the topological quantum computing device may instantiate the plurality of MZMs in a one-sided architecture or a two-sided architecture.

According to this first aspect, the processor may be configured to identify the plurality of measurement sequences that implement the logic gate at least in part by identifying one or more measurement sequences that implement the logic gate multiplied by an overall Pauli operator.

According to this first aspect, the processor may be further configured to track a Pauli gate correction to the logic gate when the topological quantum computing device implements the logic gate.

According to this first aspect, the topological quantum computing device may be configured to implement the logic gate at least in part by performing a forced measurement of a projection operator. The forced measurement may include performing a first measurement of a topological charge of the quantum state. The forced measurement may further include determining whether a result of the first measurement is a predetermined target value. When the result of the first measurement is not the predetermined target value, the forced measurement may further include resetting the quantum state and repeating the first measurement of the topological charge.

According to this first aspect, resetting the quantum state may include repeating a second measurement that was performed prior to the first measurement in the measurement sequence.

According to this first aspect, resetting the quantum state includes measuring a plurality of topological charges including a symmetric difference of a first plurality of topological charges on which the first measurement was performed and a second plurality of topological charges on which a second measurement was performed prior to the first measurement in the measurement sequence.

According to another, second aspect of the present disclosure, which is not covered by the scope of the claims, a method for performing a quantum computation is provided. The method may include identifying a plurality of measurement sequences that implement a logic gate. Each measurement sequence may include a plurality of measurements of a quantum state of a topological quantum computing device. The method may further include determining a respective estimated total resource cost of each measurement sequence of the plurality of measurement sequences. The method may further include determining a first measurement sequence that has a lowest estimated total resource cost of the plurality of measurement sequences. The method may further include implementing the logic gate at the topological quantum computing device by applying the first measurement sequence to the quantum state.

According to this second aspect, for each measurement sequence of the plurality of measurement sequences, determining the respective estimated total resource cost may include determining an estimated weighted resource cost of each measurement included in the measurement sequence. The estimated total resource cost of the measurement sequence may be determined based on the plurality of estimated weighted resource costs. The estimated weighted resource cost of each measurement may indicate an error rate of the measurement.

According to second aspect, the topological quantum computing device may include a plurality of Majorana zero modes (MZMs).

According to this second aspect, the topological quantum computing device may instantiate the plurality of MZMs in a one-sided architecture or a two-sided architecture. The topological quantum computing device may include at least one of a Majorana tetron including MZMs, a Majorana hexon including six MZMs, or a Majorana octon including eight MZMs.

According to this second aspect, identifying the plurality of measurement sequences that implement the logic gate may include identifying one or more measurement sequences that implement the logic gate multiplied by an overall Pauli operator.

According to second aspect, implementing the logic gate may include performing a forced measurement of a projection operator included in the logic gate. The forced measurement may include performing a first measurement of a topological charge of the quantum state. The forced measurement may further include determining whether a result of the first measurement is a predetermined target value. When the result of the first measurement is not the predetermined target value, the forced measurement may further include resetting the quantum state and repeating the first measurement of the topological charge.

According to another, third aspect of the present disclosure, which is not covered by the scope of the claims, a computing system is provided, including a processor configured to identify a plurality of measurement sequences that implement a logic gate. Each measurement sequence may include a plurality of measurements of a quantum state of a topological quantum computing device. The topological quantum computing device may include a Majorana hexon including six Majorana zero modes (MZMs). The processor may be further configured to determine an estimated weighted resource cost of each measurement included in each measurement sequence of the plurality of measurement sequences. For each measurement sequence, the processor may be further configured to determine the estimated total resource cost of the measurement sequence based on the estimated weighted resource costs of the measurements included in the measurement sequence. The processor may be further configured to determine a first measurement sequence that has a lowest estimated total resource cost of the plurality of measurement sequences. The topological quantum computing device may be configured to implement the logic gate by applying the first measurement sequence to the quantum state.

Claim 1:
A computing system comprising:
a topological quantum computing device (<NUM>); and
a processor (<NUM>) configured to:
identify a plurality of measurement sequences (<NUM>, <NUM>), wherein each measurement sequence of the plurality of measurement sequences (<NUM>, <NUM>) implements a same logic gate (<NUM>), and includes a plurality of measurements of a quantum state (<NUM>) of the topological quantum computing device (<NUM>);
determine a respective estimated total resource cost (<NUM>, <NUM>) of each measurement sequence of the plurality of measurement sequences (<NUM>, <NUM>); and
determine a first measurement sequence of the plurality of measurement sequences (<NUM>, <NUM>) that has a lowest estimated total resource cost,
wherein the topological quantum computing device (<NUM>) is configured to implement the logic gate by applying the first measurement sequence to the quantum state (<NUM>).