Patent Publication Number: US-2022215069-A1

Title: Cluster-state quantum computing methods and systems

Description:
RELATED APPLICATIONS 
     This application claims priority to U.S. provisional patent application No. 62/842,478, filed May 2, 2019 and titled “Continuous-Variable Quantum Computing with Photonic Cluster States”, the entirety of which is incorporated herein by reference. 
    
    
     GOVERNMENT RIGHTS 
     This invention was made with government support under Grant No. W911NF-18-1-0377, awarded by ARMY/ARO. The government has certain rights in the invention. 
    
    
     BACKGROUND 
     In measurement-based quantum computing, a quantum algorithm is implemented by performing a sequence of single-node measurements on a cluster state of qubits arranged in a square-grid topology. Gaussian cluster states may be prepared using squeezed vacuum states and linear optics, both of which are physically realizable using techniques known in the art. Although large entangled Gaussian state clusters have been experimentally demonstrated, no-go theorems show that Gaussian states alone cannot be used for universal quantum computing. To achieve universality, at least one non-Gaussian resource is required to complete the “toolkit”. 
     Examples of non-Gaussian resources that have been proposed for universal quantum computing include Gottesman-Kitaev-Preskill (GKP) states, cat states, photon number detection, and single-photon states. Although these proposed resources are mathematically elegant, many are impractical to physically implement. For example, in the Knill-Laflamme-Millburn model of quantum computing, the non-Gaussian resource is introduced by a nonlinear phase flip (i.e., a cubic phase gate). However, it is unknown how to implement such a nonlinear phase flip. In continuous variable quantum computing, the GKP model proposes the creation of a resource cluster state using momentum eigenstates and controlled-Z gates. However, momentum eigenstates correspond to nonphysical infinitely-squeezed states, and it is not known how to physically implement such states with finite squeezing. 
     SUMMARY OF THE EMBODIMENTS 
     The present embodiments include a hybrid architecture that combines continuous variable (CV) and discrete variable (DV) techniques to advantageously implement scalable, universal, CV photonic quantum computing using the currently available technologies of squeezed photon sources, photon-number-resolving detectors, and linear optics. Embodiments herein use quantum bits, or qubits, as opposed to quantum modes, or qumodes. However, these qubits are encoded in CV “cat-like” states that approximate true Schrödinger cat states. Qubits in cat-like states may form an entangled cluster state that can be advantageously used for fault-tolerant universal quantum computing without complex nonlinear phase gates. 
     The largest entangled states that have been experimentally generated with individually-addressable quantum systems are multimode squeezed states with thousands of entangled optical modes that are simultaneously available. The present embodiments may be scaled to operate with such large entangled states, and may be further combined with one-way quantum computation techniques to implement a photonic quantum computer that meets DiVincenzo criteria. 
     In embodiments, a cluster-state quantum computing method includes transforming a Gaussian graph state into a non-Gaussian percolated graph state by probabilistically subtracting one photon from each of a plurality of modes forming the Gaussian graph state. The method also includes determining cat-basis qubits of the non-Gaussian percolated graph state for which one photon was successfully subtracted from a corresponding one of the modes, and identifying in the non-Gaussian percolated graph state a renormalized graph of logical qubits connected by percolation highways. The logical qubits and percolation highways are formed from the cat-basis qubits. The method also includes outputting the renormalized graph and the non-Gaussian percolated graph state to a one-way quantum computer. 
     In embodiments, a cluster-state quantum computing system includes an array of photon subtractors configured to transform a Gaussian graph state into a non-Gaussian percolated graph state by probabilistically subtracting one photon from each of a corresponding plurality of modes forming the Gaussian graph state. Each of the photon subtractors includes a single-photon detector configured to output a detector signal. The system also includes a renormalizer configured to process the detector signal outputted by each single-photon detector to determine cat-basis qubits of the non-Gaussian percolated graph state for which one photon was successfully subtracted from a corresponding one of the modes. The renormalizer is also configured to identify in the non-Gaussian percolated graph state a renormalized graph of logical qubits connected by percolation highways, wherein the logical qubits and percolation highways are formed from the cat-basis qubits. The renormalizer is also configured to output the renormalized graph and the non-Gaussian percolated graph state to a one-way quantum computer. 
    
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
         FIG. 1  shows transformation of a Gaussian graph state into a renormalized cluster state that can be subsequently used as a quantum resource for universal quantum computation, in embodiments. 
         FIG. 2  shows a renormalized graph identifying connected qubits that form vertical percolation highways, horizontal percolation highways, and crossover qubits, in embodiments. 
         FIG. 3  is a renormalized graph similar to the renormalized graph of  FIG. 2  except that each logical qubit of the logical lattice state corresponds to one of a plurality of cluster substrates, each formed from a group of connected cat-basis qubits, in embodiments. 
         FIG. 4  is a functional diagram of a percolator that converts the Gaussian graph state into the non-Gaussian percolated graph state for subsequent processing by a one-way quantum computer, in an embodiment. 
         FIG. 5  is a functional diagram illustrating how the one-way quantum computer of  FIG. 4  can cooperate with a renormalizer to execute a quantum algorithm with the renormalized graph state, in an embodiment. 
         FIG. 6  is a flow chart of a cluster-state quantum computing method  600 . 
     
    
    
     DETAILED DESCRIPTION OF THE EMBODIMENTS 
       FIG. 1  shows transformation of a Gaussian graph state  100  into a renormalized cluster state  140  that can be subsequently used as a quantum resource for universal quantum computation. The Gaussian graph state  100  is a squeezed state containing a plurality of entangled modes  102 . The Gaussian graph state  100  is depicted in  FIG. 1  as a two-dimensional mathematical graph with nodes representing modes  102  and edges, or links, representing pair-wise entanglement  104  between neighboring modes  102 . Thus, each of the modes  102  is entangled with its nearest-neighbor modes  102 . For example, each mode  102  away from the periphery of the Gaussian graph state  100  has four nearest-neighbor modes  102 . Each of the modes  102  is orthogonal to all of the other modes  102 , and may be a classical mode or a quantum mode. In embodiments, the modes  102  are quantum electromagnetic modes arising from quantization of the electromagnetic field, and may correspond to spatial modes, temporal modes, polarization modes, frequency modes, or a combination thereof. 
     While  FIG. 1  shows the Gaussian graph state  100  represented as a square lattice having eight rows  106  and eight columns  108 , the Gaussian graph state  100  may have any number N R &gt;1 of rows  106  and any number N c &gt;1 of columns  108  without departing from the scope hereof. In other embodiments, the Gaussian graph state  100  forms an n-dimensional graph, where n&gt;2. For example, the Gaussian graph state  100  may form a three-dimensional cubic lattice, wherein each of the modes  102  away from the periphery is entangled with six nearest-neighbor modes  102 . 
     The Gaussian graph state  100  is converted into a non-Gaussian percolated graph state  120  via photon subtraction  110  of modes  102 . Photon subtraction  110  probabilistically transforms the modes  102  into cat-basis qubits  122  that collectively form a multimode cat-basis entangled state |ψ ±   . The cat-basis entangled state |Ψ ±    approximates a true multimode Greenberger-Horne-Zeilinger (GHZ) cat state |C +    of the form 
     
       
         
           
             
               
                 
                   
                     
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     where N ±  is a normalization constant and each mode of |C +    is a coherent state with complex amplitude a. The number of amplitudes α in each ket in the right side of Eqn. 1 equals the number of modes  102  forming the Gaussian graph state  100 . When |α| is small, |Ψ ±   ≠|C ±   , and thus |Ψ ±    may be used in place of |C ±    to perform universal quantum computation. 
     One aspect of the embodiments is the realization that a non-Gaussian N-mode cat-basis entangled state |Ψ ±    can be formed from the N-mode Gaussian graph state  100  by directly subtracting one photon from each of the N modes  102 . The present embodiment may advantageously achieve a higher success probability (i.e., a higher probability that one photon was successfully subtracted from each of the N modes  102 ), and a higher fidelity, as compared to the technique of subtracting N photons from a single-mode squeezed state to create a single-mode cat-basis state, and subsequently converting the single-mode cat-basis state into the N-mode cat-basis entangled state |Ψ ±    via coupling with the vacuum state in a plurality of beamsplitters. 
     In  FIG. 1 , empty circles of the percolated graph state  120  represent cat-basis qubits  122 , i.e., modes  102  that were successfully transformed into a cat-basis state due to photon subtraction  110  (i.e., one photon was subtracted from the corresponding mode  102 ). Solid circles of the percolated graph state  120  are untransformed modes  124  for which no photons were detected. The untransformed modes  124  are therefore the same as the modes  102 . 
     Renormalization  130  identifies in the percolated graph state  120  a plurality of cat-like connected qubits  142  that form a renormalized graph state  140 . Thus, the renormalized graph state  140  is a substrate of the percolated graph state  120  wherein each of connected qubits  142  is one of the cat-basis qubits  122 . Renormalization  130  generates a renormalized graph (see the renormalized graphs  200  and  300  of  FIGS. 2 and 3 , respectively) that corresponds to the percolated graph state  120  but identifies connected qubits  142  forming the renormalized graph state  140 . As described in more detail below, renormalization  130  identifies the connected qubits  142  by finding within the percolated graph state  120  a plurality of “percolation highways”, i.e., long-range, crossing, edge-disjoint, one-dimensional chains of connected qubits  142  (see percolation highways  202 ,  204  in  FIGS. 2 and 3 ). 
       FIG. 2  shows a renormalized graph  200  identifying connected qubits  142  that form vertical percolation highways  202 , horizontal percolation highways  204 , and crossover qubits  206 . The connected qubits  142  are shown in  FIG. 2  as white circles, while all other modes/qubits are shown as black circles (i.e., untransformed modes  124  of the percolated graph state  120 , and the cat-basis qubits  122  that are both excluded from the percolation highways  202 ,  204  and are not crossover qubits  206 ). For clarity, entanglement  104  is only shown in  FIG. 2  between the connected qubits  142  forming the percolation highways  202 ,  204 . While  FIG. 2  shows the percolation highways  202 ,  204  fully extending across opposite sides of the renormalized graph  200 , the percolation highways  202 ,  204  need not fully extend all the way to any side of the renormalized graph  200 . 
     In  FIG. 2 , each cross-over qubit  206  is located where one of the vertical percolation highways  202  crosses one of the horizontal percolation highways  204 . In the example of  FIG. 2 , four vertical percolation highways  202  cross four horizontal percolation highways  204  to generate sixteen cross-over qubits  206 , of which only two are indicated for clarity. However, there may be a different number of vertical percolation highways  202  and/or horizontal percolation highways  204  in the renormalized graph  200 . 
     The percolation highways  202 ,  204  may be represented as a two-dimensional logical lattice state  210  formed from logical qubits  220  connected to neighboring logical qubits  220  via logical entanglement  222 . Each of the logical qubits  220  corresponds to one of the cross-over qubits  206 , and each connection of logical entanglement  222  (i.e., each edge connecting two neighboring logical qubits  220 ) corresponds to one entanglement chain  208  of connected qubits  142  that joins a pair of neighboring cross-over qubits  206 . In the example of  FIG. 2 , there are twenty-four entanglement chains  208 , of which only one is indicated in the renormalized graph  200  for clarity. 
     After renormalization  130 , a one-way quantum computer (see one-way quantum computer  440  in  FIGS. 4 and 5 ) may use the renormalized graph state  140  by processing the percolated graph state  120  according to the renormalized graph  200 . For example, where a node of the renormalized graph  200  indicates that a corresponding mode/qubit of the percolated graph state  120  is not a connected qubit  142  (i.e., a node with a black circle in  FIG. 2 ), the quantum computer may perform a z-measurement on the mode/qubit to remove it from the percolated graph state  120 , and to remove its entanglement to neighboring modes/qubits. Where a node of the renormalized graph  200  indicates that a corresponding connected qubit  142  belongs to an entanglement chain  208 , the quantum computer may perform an x-measurement on the connected qubit  142  to remove it from the percolated graph state  120  while bridging the entanglement chain  208  (i.e., entangling the two nearest-neighbor connected qubits  142  that also belong to the entanglement chain  208 ). Where the node of the renormalized graph  200  indicates that a corresponding qubit is a cross-over qubit  206 , the quantum computer may measure the cross-over qubit  206  according to a quantum algorithm. 
       FIG. 3  is a renormalized graph  300  similar to the renormalized graph  200  of  FIG. 2  except that each logical qubit  220  of the logical lattice state  210  corresponds to one of a plurality of cluster substrates  306 , each formed from a group of connected cat-basis qubits  142 . That is, a single logical qubit  220  is represented by multiple physical qubits  142 . Each cluster substrate  306  may be used to implement one logical qubit  220  with error correction to achieve fault-tolerant quantum computing. Examples of error-correction methods that may be used with cluster substrates  306  are known in the art. For clarity in  FIG. 3 , only three cluster substrates  306  are indicated. However, one cluster substrate  306  may be formed where one vertical percolation highway  202  crosses one horizontal percolation highway  204 . 
     In the preceding discussion, the cat-basis qubits  122  and connected qubits  142  are represented in the cat basis. However, the present embodiments may be used with qubits in a GKP basis, or another type of hybrid CV non-Gaussian orthogonal qubit basis without departing from the scope hereof 
     Physical Implementation 
       FIG. 4  is a functional diagram of a percolator  400  that converts the Gaussian graph state  100  into the non-Gaussian percolated graph state  120  for subsequent processing by a one-way quantum computer  450 . In  FIG. 4 , the Gaussian graph state  100  is physically implemented as a plurality of spatially-separated optical beams, each corresponding to one row  106  of the Gaussian graph state  100 . The columns  108  of the Gaussian graph state  100  correspond to different times separated by a time interval Δt. Thus, the modes  102  are spatial-temporal modes, and the Gaussian graph state  100  is processed one column at a time over a duration of N c ×Δt, where N c  is the number of the columns  108 . Although entanglement  104  between the rows  106  is not shown in  FIG. 4  for clarity, it is implied that the modes  102  are entangled between the rows  106  according to the Gaussian graph state  100  of  FIG. 1 . 
     The percolator  400  includes an array of photon subtractors  408  that probabilistically transform each mode  102  into a cat-basis qubit  122  by subtracting one photon from said each mode  102 . The percolator  400  also includes an array of optical delays  414  and an array of PNR photodetectors  410 . There are N photon subtractors  408  and N optical delays  414 , where N is the number of rows  106  (i.e., the number of spatially-separated optical beams being processed). Thus, each row  106  passes through a corresponding one of the photon subtractors  408  and one of the optical delays  414 . With this architecture, the percolator  400  processes the N rows  106  in parallel. 
     In  FIG. 4 , a first photon subtractor  408 ( 1 ) includes a first beamsplitter  404 ( 1 ) with a high transmission (e.g., 98%) and a first PNR detector  410 ( 1 ) coupled to a first output port of first beamsplitter  404 ( 1 ). A first row  106 ( 1 ) of the Gaussian graph state  100  is coupled to a first input port of the first beamsplitter  404 ( 1 ), and the vacuum state  10 ) is coupled to a second input port of the first beamsplitter  404 ( 1 ). The first PNR detector  410 ( 1 ) outputs a first detector signal  412 ( 1 ) in response to photons detected in the first output port. The cat-basis qubits  122  and untransformed modes  124  are outputted from a second output port of the first beamsplitter  404 ( 1 ) as a first output stream  406 ( 1 ) to form a corresponding row of the non-Gaussian percolated graph state  120 . Although not shown in  FIG. 4  for clarity, it is implied that the cat-basis qubits  122  and untransformed modes  124  of one output stream  406  are entangled with qubits/modes in neighboring output streams  406  according to the non-Gaussian percolated graph state  120  of  FIG. 1 . 
     The modes  102  of the first row  106 ( 1 ) are sequentially inputted to the first photon subtractor  408 ( 1 ). Transformation of each mode  102  into a cat-basis qubit  122  is conditioned upon detection of one photon by the first PNR detector  410 ( 1 ), and thus is a probabilistic process. For example,  FIG. 4  shows one cat-basis qubit  122  exiting the first photon subtractor  408 ( 1 ). Simultaneously, the first detector signal  412 ( 1 ) contains a peak  416  indicating that one photon was detected by the first PNR detector  410 ( 1 ). In this case, the peak  416  indicates that a corresponding mode  102  of the first row  106 ( 1 ) was successfully transformed into the cat-basis qubit  122 . 
       FIG. 4  also shows one untransformed mode  124  exiting the first photon subtractor  408 ( 1 ). Simultaneously, the first detector signal  412 ( 1 ) contains no peak, i.e., no photons were detected by the first detector  410 ( 1 ). This absence of a peak  416  indicates that a corresponding mode  102  of the first row  106 ( 1 ) was not successfully transformed into a cat-basis qubit. Equivalently, the photon subtractor  408 ( 1 ) subtracted zero photons from the corresponding mode  102 , and thus remains in a Gaussian state. 
     The percolator  400  also includes a second photon subtractor  408 ( 2 ) that operates similarly to the first photon subtractor  408 ( 1 ). Specifically, the second photon subtractor  408 ( 2 ) includes a second beamsplitter  404 ( 2 ) and a second PNR detector  410 ( 2 ) that cooperate to transform a second row  106 ( 2 ) of the Gaussian graph state  100  into a second output stream  406 ( 2 ) and a corresponding second detector signal  412 ( 2 ). The percolator  400  may contain additional photon subtractors  408 , as needed to process all of the rows  106  of the Gaussian graph state  100 . Thus, while  FIG. 4  shows the percolator  400  with three photon subtractors  408  processing three rows  106 , the percolator  400  may include a different number of photon subtractors  408  without departing from the scope hereof. 
     The photon subtractors  408  transform each mode  102  into a cat-basis qubit  122  with a success probability p. When p is close to 1, almost every mode  102  is successfully transformed into the cat-basis qubit  122 , in which case the percolated graph state  120  forms several percolation highways  202 ,  204 , and a renormalized graph state  140  can be identified with high probability. However, when the probability p falls below a percolation threshold, there are too few cat-basis qubits  122  to form any percolation highways  202 ,  204 , in which case the percolated graph state  120  contains insufficient non-Gaussian resources for universal quantum computing. The percolator  400  and/or Gaussian graph state  100  may be configured to ensure that p is greater than the percolation threshold. For example, squeezing of the Gaussian graph state  100  and/or reflectivity of the beamsplitters  404  may be selected to achieve a desired probability p. 
     The percolation threshold may be calculated for different types of cluster states. Embodiments herein implement site percolation by considering the untransformed modes  124  as having been “removed” from the Gaussian graph state  100 . This contrasts with bond percolation, in which the edges (i.e., entanglement  104 ) between nodes (i.e., modes  102 ) are “removed”. For the case of bond percolation, example values of the percolation threshold are known in the art. 
     The optical delays  414  delay the output streams  406  so that the photon subtractors  408  can process a sequence of M columns of the Gaussian graph state  100  before the first column of the sequence is processed by the one-way quantum computer  440 . Thus, the optical delays  414  delay the output streams  406  by M×Δt. This delay is selected based on a desired size of the renormalized graph state  140  and/or logical lattice state  210 . Each of the optical delays  414  may be an optical fiber, a folded optical delay line, or another type of optical delay system known in the art. 
       FIG. 5  is a functional diagram illustrating how the one-way quantum computer  440  of  FIG. 4  can cooperate with a renormalizer  502  to execute a quantum algorithm  510  with the renormalized graph state  140 . The renormalizer  502  receives the detector signals  412  from the photon subtractors  408  of  FIG. 4 , processes the detector signals  412  to construct the graph  200  of the percolated graph state  120 , and renormalizes the graph  200  to identify the renormalized graph state  140  (i.e., percolation highways  202  and  204 , crossover qubits  206 , entanglement chains  208 , etc.). The renormalizer  502  outputs the renormalized graph  200  to a controller  506  of the one-way quantum computer  440 . The controller  506  returns an output  520  when execution of the quantum algorithm  510  has finished. 
     The one-way quantum computer  440  also includes homodyne detectors  530  that detect the modes  102  of the output streams  406  (after the optical delay  414 , as shown in  FIG. 4 ). Each of the homodyne detectors  530  includes a variable phase shifter (not shown) that may be controlled to detect a qubit in a selected basis. The selected bases are programmed by the controller  506  via control lines  512 . Data outputted by the homodyne detectors  530  is communicated back to the controller  506  via data lines  514 , where the controller  506  uses the received data to select new bases for the next qubit measurements. The controller  506  also selects the bases according to the renormalized graph  200  so that quantum information only flows along the renormalized graph state  140 . 
     In the examples of  FIGS. 4 and 5 , the Gaussian graph state  100  is implemented as a two-dimensional cluster state of spatio-temporal modes  102 . This cluster state is also known as a time-domain multiplexed cluster state. With this implementation, the Gaussian graph state  100  may be generated from an array of squeezed-light generators, or “squeezers”, each outputting a single-mode squeezed-vacuum pulse-train. The outputs of the squeezers are spatially-separated optical beams that may be processed in parallel. The squeezers may be operated synchronously such that all the squeezers output one mode simultaneously. These modes may be entangled to each other using a network of beamsplitters, thereby creating vertical edges in one column  108  of the Gaussian graph state  100 . Modes may be further entangled to each other using optical time delays in the network of beamsplitters, thereby creating horizontal edges in the Gaussian graph state  100 . 
     In one embodiment, each squeezer is an optical parametric oscillator (OPO). For example, the time-domain multiplexed Gaussian graph state  100  may be created from four OPOs and a network of five beamsplitters and two optical time delays, as known in the art. In this reference, the Gaussian graph state  100  is encoded onto four optical beams, each coupled into one photon subtractor  408  of  FIG. 4 . However, the time-domain multiplexed Gaussian graph state  100  may be alternatively created with a different number n of similarly-configured OPOs, wherein the Gaussian graph state  100  is encoded into n optical beams subsequently processed by n corresponding photon subtractors  408 . In another embodiment, each of the squeezers is an optical parametric amplifier (OPA). In another embodiment, each of the squeezers is a nano-photonic squeezer. The nano-photonic squeezer may be based on an integrated periodically-poled nonlinear crystal (e.g., PPLN, PPKTP) or on a ring resonator (e.g., using SiN or AlN). 
     In some embodiments, the array of squeezers is fabricated on a single photonic integrated circuit (PIC). The beamsplitter network and/or optical time delays used to entangle the outputs of the squeezers may also be incorporated on the PIC. The beamsplitters may be variable beamsplitters that can be controlled to correct for manufacturing imperfections and/or implement protocols that engineer the resulting multimode Gaussian cluster state for one-way quantum computing. 
     In another embodiment, each of the squeezers is powered by a pump laser beam with a controllable pump level (e.g., intensity). The pump levels of the pump laser beams are controlled such that the array of squeezers directly generates the Gaussian cluster state  100 , thereby eliminating the need for the beamsplitter network. 
     In other embodiments, the Gaussian graph state  100  is implemented as a cluster state of entangled frequency modes  102  having the same spatial, temporal, and polarization modes. These frequency modes may be generated, for example, by a quantum optical frequency comb (QOFC), i.e., a single OPO driven by a multifrequency pump and enclosed in an optical cavity forming a comb-like structure of adjacent optical resonances. QOFCs have been used to generate multipartite entanglement of thousands of quantum modes each uniquely identified by the frequency of the corresponding optical resonance. The output of the QOFC is a single optical beam containing pairwise-entangled frequency modes  102  (i.e., frequency-staggered EPR pairs). A subsequent beamsplitter network completes the entanglement between EPR pairs to generate Gaussian graph state  100 . To use the QOFC output with the percolator  300 , frequency-domain beamsplitters and PNR detectors are needed such that each of the frequency modes  102  can be processed individually. Alternatively, the frequency modes  102  may be spatially separated, for example, with a virtually-imaged phased array, prism, or other type of dispersive optical element. 
     When the Gaussian graph state  100  is implemented as a cluster state of N entangled frequency modes  102 , all N frequency modes  102  may be generated simultaneously. In one embodiment, all N frequency modes  102  are dispersed into spatially-separated beams prior to photon subtraction. In this embodiment, N photon subtractors  408  process N frequency modes  102  simultaneously, wherein each output stream  406  contains only one mode (i.e., either one cat-basis qubit  124  or one untransformed mode  124 ). In this embodiment, the optical delays  414  are configured with different delays such that some of the resulting cat-basis qubits  124  are processed by the one-way quantum computer  440  prior to other cat-basis qubits  124 , thereby allowing the one-way quantum computer  440  to process the cat-basis qubits  124  in a time-multiplexed way. 
     In one embodiment, all N frequency modes  102  are photon subtracted while remaining in the single beam outputted by the QOFC. In this embodiment, only one photon-subtracting beamsplitter  404  is needed. To identify the success of one-photon subtraction for each of the N frequency modes  102 , the first output port of the beamsplitter  404  may be spatially dispersed into N optical beams detected by N corresponding PNR detectors  410 . The spatial dispersion may be achieved with a virtually-imaged phased array, prism, or other type of dispersive optical element. The cat-basis qubits  122  and untransformed modes  124  form one output stream  406 . 
     In another embodiment, the Gaussian graph state  100  is implemented as a cluster state of entangled time-frequency modes  102  having the same spatial and polarization modes. In this implementation, each row  106  of the Gaussian graph state  100  corresponds to a single frequency, and the columns  108  correspond to different times. Time-frequency modes  102  may be generated from a QOFC by operating the QOFC in pulsed mode, and different temporal modes may be entangled using a beamsplitter network with optical delays (thereby generating horizontal edges in the Gaussian graph state  100 , as depicted in  FIG. 1 ). In this implementation, the one-way quantum computer  440  processes the modes  102  at different times (i.e., one column at a time), thereby operating in a time-multiplexed way without varying optical time delays. 
     In one embodiment, the QOFC is powered by a multi-frequency pump laser beam where each frequency component has a controllable pump level (e.g., intensity). The pump levels are controlled such that the QOFC directly generates the Gaussian cluster state  100 , thereby eliminating the need for any beamsplitter network after the QOFC. 
     METHOD EMBODIMENTS 
       FIG. 6  is a flow chart of a cluster-state quantum computing method  600 . In the block  602 , a Gaussian graph state is transformed into a non-Gaussian percolated graph state by probabilistically subtracting one photon from each of a plurality of modes forming the Gaussian graph state. In one example of the block  602 ,  FIG. 1  shows how the Gaussian graph state  100  may be transformed into the non-Gaussian percolated graph state  120  via photon subtraction  110  of modes  102 . In another example of the block  602 , the percolator  400  of  FIG. 4  uses the array of photon subtractors  408  to transform the Gaussian graph state  100  into the non-Gaussian percolated graph state  120 . 
     In the block  604  of the method  600 , cat-basis qubits of the non-Gaussian percolated graph state are determined for which one photon was successfully subtracted from a corresponding one of the modes. In one example of the block  604 , photon subtraction  110  probabilistically transforms the modes  102  into cat-basis qubits  122 , as shown in  FIG. 1 . In another example of the block  604 , The renormalizer  502  of  FIG. 4  receives the detector signals  412  from the photon subtractors  408  of  FIG. 4  and processes the detector signals  412  to construct the graph  200  of the percolated graph state  120 . 
     In the block  606  of the method  600 , a renormalized graph of logical qubits connected by percolation highways is identified in the non-Gaussian percolated graph state. The logical qubits and percolation highways are formed from the cat-basis qubits. In one example of the block  606 ,  FIG. 2  shows the renormalized graph  200  identifying connected qubits  142  that form vertical percolation highways  202 , horizontal percolation highways  204 , and crossover qubits  206 . In another example of the block  606 , the renormalizer  502  of  FIG. 5  renormalizes the graph  200  to identify the renormalized graph state  140 . 
     In the block  608  of the method  600 , the renormalized graph and the non-Gaussian percolated graph state are outputted to a one-way quantum computer. In one example of the block  608 , the one-way quantum computer  440  of  FIG. 4  processes the output streams  406 . In another example of the block  608 , the renormalizer  502  outputs the renormalized graph  200  to a controller  506  of the one-way quantum computer  440 . 
     In some embodiments, the method  600  includes the block  610 , in which the one-way quantum computer processes the non-Gaussian percolated graph state according to the renormalized graph to implement a quantum computing algorithm. In one example of the block  610 , the one-way quantum computer  440  of  FIG. 4  returns the output  520  when execution of the quantum algorithm  510  has finished. 
     Combination of Features 
     Features described above as well as those claimed below may be combined in various ways without departing from the scope hereof. The following examples illustrate possible, non-limiting combinations of features and embodiments described above. It should be clear that other changes and modifications may be made to the present embodiments without departing from the spirit and scope of this invention: 
     (A1) A cluster-state quantum computing method may include transforming a Gaussian graph state into a non-Gaussian percolated graph state by probabilistically subtracting one photon from each of a plurality of modes forming the Gaussian graph state. The method may also include determining cat-basis qubits of the non-Gaussian percolated graph state for which one photon was successfully subtracted from a corresponding one of the modes, and identifying in the non-Gaussian percolated graph state a renormalized graph of logical qubits connected by percolation highways. The logical qubits and percolation highways may be formed from the cat-basis qubits. The method may also include outputting the renormalized graph and the non-Gaussian percolated graph state to a one-way quantum computer. 
     (A2) In the method denoted (A1), the cluster-state quantum computing method may include processing, with the one-way quantum computer, the non-Gaussian percolated graph state according to the renormalized graph to implement a quantum computing algorithm. 
     (A3) In either of the methods denoted (A1) and (A2), said identifying may include locating connected qubits, of the cat-basis qubits, that form the percolation highways in the non-Gaussian percolated graph state, forming the logical qubits from at least some of the connected qubits, and forming, from the percolation highways, entanglement chains that link the logical qubits. 
     (A4) In any one of the methods denoted (A1) to (A3), said transforming the Gaussian graph state into the non-Gaussian percolated graph state may include parallelly processing a plurality of spatially-separated registers that form the Gaussian graph state. 
     (A5) In the method denoted (A4), the cluster-state quantum computing method may include, for each of the registers, subtracting one photon from each mode of the Gaussian graph state by: (i) entangling said each mode with a vacuum state by coupling said each mode to a first input port of a beamsplitter and coupling the vacuum state to a second input port of the beamsplitter, and (ii) measuring, with a photodetector at a first output port of the beamsplitter, the one photon when successfully subtracted from said each mode. Said determining the cat-basis qubits may include labeling said each mode as one of the cat-basis qubits based on an output of the photodetector. 
     (A6) In any one of the methods denoted (A1) to (A5), the cluster-state quantum computing method may include creating the Gaussian graph state by generating a multimode squeezed vacuum state that forms the plurality of modes. 
     (A7) In the method denoted (A6), said generating the multimode squeezed vacuum state may use a quantum optical frequency comb. Each of the plurality of modes may correspond to one of a plurality of frequencies of the quantum optical frequency comb. 
     (A8) In the method denoted (A7), the cluster-state quantum computing method may include dispersing the multimode squeezed vacuum state to spatially separate the plurality of modes. 
     (B1) A cluster-state quantum computing system may include an array of photon subtractors configured to transform a Gaussian graph state into a non-Gaussian percolated graph state by probabilistically subtracting one photon from each of a corresponding plurality of modes forming the Gaussian graph state. Each of the photon subtractors may include a single-photon detector configured to output a detector signal. The system may also include a renormalizer configured to process the detector signal outputted by each single-photon detector to determine cat-basis qubits of the non-Gaussian percolated graph state for which one photon was successfully subtracted from a corresponding one of the modes. The renormalizer may also be configured to identify in the non-Gaussian percolated graph state a renormalized graph of logical qubits connected by percolation highways, wherein the logical qubits and percolation highways are formed from the cat-basis qubits. The renormalizer may also be configured to output the renormalized graph and the non-Gaussian percolated graph state to a one-way quantum computer. 
     (B2) In the system denoted (B1), the cluster-state quantum computing system may include the one-way quantum computer. The one-way quantum computer may be configured to process the non-Gaussian percolated graph state according to the renormalized graph to implement a quantum computing algorithm. 
     (B3) In the system denoted (B2), the one-way quantum computer may include an array of homodyne detectors configured to detect the modes. 
     (B4) In any one of the systems denoted (B1) to (B3), the renormalizer may be configured to identify the renormalized graph by (i) locating connected qubits, of the cat-basis qubits, that form the percolation highways in the non-Gaussian percolated graph state, (ii) forming the logical qubits from at least some of the connected qubits, and (iii) forming, from the percolation highways, entanglement chains that link the logical qubits. 
     (B4) In any one of the systems denoted (B1) to (B4), the renormalizer may be configured to transform the Gaussian graph state into the non-Gaussian percolated graph state by parallelly processing, with the array of photon subtractors, a corresponding array of spatially-separated registers that form the Gaussian graph state. 
     (B5) In the system denoted (B4), each of the photon subtractors may include a beamsplitter configured to entangle the corresponding mode with a vacuum state by coupling the corresponding mode to a first input port of the beamsplitter and coupling the vacuum state to a second input port of the beamsplitter. 
     (B6) In either one of the systems denoted (B4) and (B5), the cluster-state quantum computing system may include an optical delay for each of the spatially-separated registers. 
     (B7) In any one of the systems denoted (B4) to (B6), the cluster-state quantum computing system may include an array of squeezed-light generators, wherein each of the squeezed-light generators outputs a single-mode squeezed-vacuum pulse-train into a corresponding one of the array of photon subtractors. 
     (B8) In the system denoted (B7), each of the squeezed-light generators may be an optical parametric oscillator. 
     (B9) In either one of the systems denoted (B7) and (B8), the array of squeezed-light generators may be configured to operate synchronously. 
     (B10) In any one of the systems denoted (B7) to (B9), the cluster-state quantum computing system may include a network of beamsplitters configured to entangle the single-mode squeezed-vacuum pulse-train outputted by each of the squeezed-light generators. 
     (B11) In any one of the systems denoted (B1) to (B9), the cluster-state quantum computing system may include a quantum optical frequency comb configured to generate a multimode squeezed vacuum state that forms the plurality of modes of the Gaussian graph state. 
     Changes may be made in the above methods and systems without departing from the scope hereof. It should thus be noted that the matter contained in the above description or shown in the accompanying drawings should be interpreted as illustrative and not in a limiting sense. The following claims are intended to cover all generic and specific features described herein, as well as all statements of the scope of the present method and system, which, as a matter of language, might be said to fall therebetween.