Boosted linear-optical measurements on single-rail qubits with unentangled ancillas

Bo osted linear-optical measuremen ts on single-rail qubits with unen tangled ancillas Aqil Sa jjad, 1, 2 , ∗ Isac k P adilla, 1 , † and Saik at Guha 1, 2 , ‡ 1 Col le ge of Optic al Scienc es, University of Arizona, T ucson AZ 85721 2 Dep artment of Ele ctric al and Computer Engine ering, University of Maryland, Col le ge Park MD 20742 An y quantum state of the radiation ﬁeld, sliced in small non-ov erlapping space-time bins is a collection of single-rail qubits, each spanning the v acuum and single-photon F ock state of a mode. Quan tum logic on these qubits would enable arbitrary measurements on information-b earing ligh t, but is hard due to the lack of strong nonlinearities. With unentangled ancilla single-rail qubits, an 8-p ort in terferometer and photon detection, we sho w any single-rail qubit measuremen t in the X Y Blo c h plane is realizable with success probability 147 / 256, which beats the prior-known 1 / 2 limit. Intr o duction. —There are many enco dings of a qubit in to b osonic mo des, suc h as the dual-rail, single-rail, GKP , and the cat-basis enco ding, eac h with its o wn pros and cons in terms of ease of preparation, resilience to photon loss [ 1 ], quantum comm unications capacity [ 2 ], in terfacing with matter quantum memories [ 3 , 4 ], and the prospects of all-optical univ ersal quan tum logic and measuremen ts [ 5 ]. Photonic quantum information pro- cessing is the pursuit of optimal means of enco ding, pro- cessing and measurement of information-b earing light so as to maximize the eﬃciency of the information pro cess- ing task at hand, b e it maximizing communications ca- pacit y [ 6 ], estimation precision of an entanglemen t-based photonic sensor [ 7 , 8 ], or the resolution of an optical imaging system [ 9 , 10 ]. Prescriptions of optimal mea- suremen ts typically arise from quantum information the- ory , and app ear in the form of abstract operator rep- resen tations that are hard to translate in to a blueprint of a readily realizable system in the mo dern quantum- optics lab oratory . One wa y to address this is to slice the information-b earing optical pulse or wa v eform into small bins of non-ov erlapping space-time mo des, since no mat- ter the kind of ligh t inv olv ed, e.g., coheren t state, mul- timo de squeezed state, thermal state or other, one can arrange the slicing to b e ﬁne enough such that eac h mo de is a single-rail photonic qubit, i.e., the qubit enco ding in the span of v acuum | 0 ⟩ , and a single photon F o c k state | 1 ⟩ of a mo de. A t this p oint, an y measurement on the original information-b earing light translates in to an in- struction set on a universal quan tum computer acting on single-rail qubits. Single-rail qubits also appear naturally in extremely lo w photon ﬂux scenarios, e.g., the weak co- heren t state pulses on the receiver end of a deep-space lasercom [ 6 ] or con tinuous-v ariable quantum-k ey distri- bution [ 11 ], or the weak m ultimo de thermal ligh t origi- nating from an astronomical source that is w ell appro x- imated b y one photon spread across a large orthogonal temp oral mo de span suc h that each mo de is a single-rail qubit [ 12 – 14 ]. Single-rail qubits are also the most natural mediator for entangling emissive atomic qubits [ 15 , 16 ]. Unfortunately , single-rail qubits are very diﬃcult to w ork with since unitary rotations in the span of | 0 ⟩ and | 1 ⟩ require highly non-linear op erations, resulting in all kno wn quantum gate prescriptions on them b eing non- deterministic [ 5 , 17 ]. Single photon detection pro vides a natural Z basis measuremen t. But measuring a single- rail qubit so as to discriminate b et ween the X basis eigen- states |±⟩ ≡ ( | 0 ⟩ ± | 1 ⟩ ) / √ 2, or more generally , the |± ϕ ⟩ ≡ 1 √ 2 ( | 0 ⟩ ± e iϕ | 1 ⟩ ) (1) states along an arbitrary azimuthal angle ϕ on the X Y plane of the qubit Bloch sphere is a ma jor c hallenge. In the same spirit, w e only know ho w to apply a non- deterministic Hadamard gate [ 5 ], which prev ents us from carrying out a deterministic X basis measuremen t by ro- tating the single-rail qubit to the Z basis. T o b ypass these issues, one could consider transfer- ring the quantum state of single-rail qubits on to opti- cally activ e quantum-logic-capable atomic memories suc h as color cen ters [ 16 , 18 ] or trapp ed ions [ 19 ], and then apply the quantum logic gates in the latter domain. This philosophy has b een explored for instance to derive prescriptions for quantum-optimal laser-ligh t discrimina- tion [ 20 ], joint-detection receiv ers for deep-space laser- ligh t communications [ 21 – 23 ], and long-base imaging telescop es [ 13 , 14 , 24 – 26 ]. The well-kno wn telep ortation- based state transfer ent ails ﬁrst entangling each single- rail photonic qubit with an atomic memory qubit (log- ical | 0 ⟩ and | 1 ⟩ ≡ ( | ↑⟩ ± | ↓⟩ ) / √ 2) by applying a con- trolled X (CNOT) gate, follo wed by measuring out the photonic qubit in the X basis, and applying a logical Z gate on the memory qubit if the measurement outcome is |−⟩ [ 27 ]. The latter step of single-qubit spin control is easy on most atomic qubits. The CNOT gate is realiz- able by reﬂecting the single-rail qubit oﬀ a ca vity-coupled atomic spin qubit in the strong coupling regime that im- parts an optical phase when a photon interacts with the atomic qubit in the | ↓⟩ state [ 28 ], whic h has b een ex- p erimen tally realized [ 29 ]. This mak es the single-rail X basis measuremen t the only b ottleneck in the aforesaid state transfer procedure. Being able to optically realize an X measurement on single-rail qubits deterministically , therefore, could op en the do or to many applications. 2 A 50% success rate for an X basis measuremen t on the single-rail qubit | ψ ⟩ can b e ac hieved by mixing | ψ ⟩ with a single-rail | + ⟩ ancilla in a balanced b eam split- ter, and measuring the photon num b ers at the t wo out- puts. This follo ws from the in timate relationship betw een measuring X on a single-rail qubit and that of the 50% success-rate Barrett-Kok partial Bell-state measurement on a pair of single-rail qubits [ 30 ], as noted in [ 14 ]. If w e detect no photons in b oth output ports, or tw o in one output and none in the other, then it means pick- ing out the | 0 ⟩ or | 1 ⟩ in our single-rail input | ψ ⟩ , hence measuring it in the Z basis and failing with the X -basis measuremen t. How ever, obtaining a single photon in one of the t w o outputs and none in the other represents a successful X basis measuremen t. This works b ecause if w e only obtain a total of 1 photon in the outputs, we cannot tell whether it came from | ψ ⟩ or the ancilla in the | + ⟩ state, hence scrambling the information ab out where the photon originated from. The b eam splitter giv es the sum of the single-photon terms in | ψ ⟩ and the ancilla in one output, and the diﬀerence in the other. With equal amplitudes for the zero and single-photon terms in | + ⟩ and |−⟩ , a single photon in the output with constructive (destructiv e) interference signiﬁes | ψ ⟩ b eing measured in the X basis and found to b e in | + ⟩ ( |−⟩ ). Note that this p erfectly constructive and destructiv e in terference only arises with a | + ⟩ or |−⟩ single-rail ancilla. It turns out ho wev er that even with a far-easier-to- pro duce coheren t state ancilla, with amplitude α = ± 1, one yields a success rate of ab out 41 . 58% for an ideal X -basis measurement. This is signiﬁcan t, since with- out ancilla assistance, X -basis measurement on single- rail qubits is not known to w ork, and the preparation of |±⟩ ancilla states is exp erimentally challenging (see Sup- plemen tary Material for further discussion.) It w as sug- gested in Ref. [ 14 ] that one could extend the | + ⟩ ancilla b oosted sc heme to a fully deterministic X measurement b y b etter hiding the whic h-path information of the pho- ton b y sending it through a m ultiple-p ort beam splitter circuit suc h as the Quantum F ourier T ransform (QFT) unitary , and employing a large num b er of | + ⟩ ancillas. In this pap er, we in vestigate a scheme for b oosting the success rate for a single-rail X basis measurement b y in- terfering the single-rail qubit | ψ ⟩ to be measured, with n − 1 unen tangled single-rail ancillas prepared in the | + ⟩ state, in an n -mo de linear in terferometer U , follow ed by photon detection (see Fig. 1 (a)). W e then extend this to measuring | ψ ⟩ in the general |± ϕ ⟩ basis along any ar- bitrary direction in the X Y plane of the Blo ch sphere b y employing a collection of | + ϕ ⟩ ancilla states. F or U , w e consider b oth the n -port Quan tum F ourier T ransform (QFT), as well as the n = 2 k ( k ∈ Z + ) mo de Hadamard unitary [ 32 , 33 ]. The p o wer-of-2 Hadamard unitaries admit a particularly simple implementation in terms of n log 2 ( n ) / 2 50-50 b eam splitters, termed the ‘Green Ma- c hine’ (GM) [ 6 ], without requiring a collection of complex Ancilla: Input: AAACHnicbVDLSgNBEJz1GeMr6tHLYBD0EnbF11H04jGCeUA2htlJbzI4M7vOzAphs1/ixV/x4kERwZP+jZNkD2osaCiquunuCmLOtHHdL2dmdm5+YbGwVFxeWV1bL21s1nWUKAo1GvFINQOigTMJNcMMh2asgIiAQyO4vRj5jXtQmkXy2gxiaAvSkyxklBgrdUpHfqgITb0s9fWdMulBlu0NXV8R2eOA/VhguEmZH/dZhoderu93SmW34o6Bp4mXkzLKUe2UPvxuRBMB0lBOtG55bmzaKVGGUQ5Z0U80xITekh60LJVEgG6n4/cyvGuVLg4jZUsaPFZ/TqREaD0Qge0UxPT1X28k/ue1EhOetlMm48SApJNFYcKxifAoK9xlCqjhA0sIVczeimmf2LyMTbRoQ/D+vjxN6gcV77hydHVYPjvP4yigbbSD9pCHTtAZukRVVEMUPaAn9IJenUfn2Xlz3ietM04+s4V+wfn8BsPlot4= 1 → 2 ( | 0 ↑ ± e i ω | 1 ↑ ) 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 ! 1 → 2 ( | 0 ↑ + e i ω | 1 ↑ ) " → 7 AAACCnicbVDLSsNAFJ3UV62vqEs30SIIaknE17LoxmUF+4AmhMn0th06mYSZiVDSrt34K25cKOLWL3Dn3zhts6itBy6cOede5t4TxIxKZds/Rm5hcWl5Jb9aWFvf2Nwyt3dqMkoEgSqJWCQaAZbAKIeqoopBIxaAw4BBPejdjvz6IwhJI/6g+jF4Ie5w2qYEKy355r6bDo59N+5SV2DeYXAyOJ1+ukPfLNolewxrnjgZKaIMFd/8dlsRSULgijAsZdOxY+WlWChKGAwLbiIhxqSHO9DUlOMQpJeOTxlah1ppWe1I6OLKGqvTEykOpeyHge4MserKWW8k/uc1E9W+9lLK40QBJ5OP2gmzVGSNcrFaVABRrK8JJoLqXS3SxQITpdMr6BCc2ZPnSe2s5FyWLu7Pi+WbLI482kMH6Ag56AqV0R2qoCoi6Am9oDf0bjwbr8aH8TlpzRnZzC76A+PrF4H8ms0= {| + ω → , | ↑ ω → } photon counting (a) (b) PS = 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 ! e i ω 0 0 e → i ω " AAAB83icbVDLSgNBEJyNrxhfUY9eBoPgKe4GXxch6MVjBPOA7BJmJ7PJkNnZYaZXCEt+w4sHRbz6M978GyfJHjSxoKGo6qa7K1SCG3Ddb6ewsrq2vlHcLG1t7+zulfcPWiZJNWVNmohEd0JimOCSNYGDYB2lGYlDwdrh6G7qt5+YNjyRjzBWLIjJQPKIUwJW8n0YMiA3vuJntV654lbdGfAy8XJSQTkavfKX309oGjMJVBBjup6rIMiIBk4Fm5T81DBF6IgMWNdSSWJmgmx28wSfWKWPo0TbkoBn6u+JjMTGjOPQdsYEhmbRm4r/ed0Uousg41KlwCSdL4pSgSHB0wBwn2tGQYwtIVRzeyumQ6IJBRtTyYbgLb68TFq1qndZvXg4r9Rv8ziK6Agdo1PkoStUR/eogZqIIoWe0St6c1LnxXl3PuatBSefOUR/4Hz+AFyPkUE= ω = ε / 2 AAAB9XicbVDJSgNBEK2JW4xb1KOXxiB4ijOiMRch6MVjBLNAZgw9nZ6kSc9Cd40ShvyHFw+KePVfvPk3dpaDRh8UPN6roqqen0ih0ba/rNzS8srqWn69sLG5tb1T3N1r6jhVjDdYLGPV9qnmUkS8gQIlbyeK09CXvOUPryd+64ErLeLoDkcJ90Laj0QgGEUj3bs44EgvnYqbiJNqt1iyy/YU5C9x5qQEc9S7xU+3F7M05BEySbXuOHaCXkYVCib5uOCmmieUDWmfdwyNaMi1l02vHpMjo/RIECtTEZKp+nMio6HWo9A3nSHFgV70JuJ/XifFoOplIkpS5BGbLQpSSTAmkwhITyjOUI4MoUwJcythA6ooQxNUwYTgLL78lzRPy06lfH57VqpdzePIwwEcwjE4cAE1uIE6NICBgid4gVfr0Xq23qz3WWvOms/swy9YH99TWpHC ω = 16 ε / 8 AAAB9XicbVDJSgNBEO2JW4xb1KOXxiB4ijMSNRch6MVjBLNAZgw9nZqkSc9Cd40ShvyHFw+KePVfvPk3dpaDRh8UPN6roqqen0ih0ba/rNzS8srqWn69sLG5tb1T3N1r6jhVHBo8lrFq+0yDFBE0UKCEdqKAhb6Elj+8nvitB1BaxNEdjhLwQtaPRCA4QyPduzgAZJdOxU3ESbVbLNllewr6lzhzUiJz1LvFT7cX8zSECLlkWnccO0EvYwoFlzAuuKmGhPEh60PH0IiFoL1sevWYHhmlR4NYmYqQTtWfExkLtR6FvukMGQ70ojcR//M6KQZVLxNRkiJEfLYoSCXFmE4ioD2hgKMcGcK4EuZWygdMMY4mqIIJwVl8+S9pnpad8/LZbaVUu5rHkScH5JAcE4dckBq5IXXSIJwo8kReyKv1aD1bb9b7rDVnzWf2yS9YH99QSJHA ω = 14 ε / 8 AAAB9XicbVDJSgNBEK2JW4xb1KOXxiB4ijNiNBch6MVjBLNAZgw9nZ6kSc9Cd40ShvyHFw+KePVfvPk3dpaDRh8UPN6roqqen0ih0ba/rNzS8srqWn69sLG5tb1T3N1r6jhVjDdYLGPV9qnmUkS8gQIlbyeK09CXvOUPryd+64ErLeLoDkcJ90Laj0QgGEUj3bs44EgvnYqbiJNqt1iyy/YU5C9x5qQEc9S7xU+3F7M05BEySbXuOHaCXkYVCib5uOCmmieUDWmfdwyNaMi1l02vHpMjo/RIECtTEZKp+nMio6HWo9A3nSHFgV70JuJ/XifFoOplIkpS5BGbLQpSSTAmkwhITyjOUI4MoUwJcythA6ooQxNUwYTgLL78lzRPy855uXJ7VqpdzePIwwEcwjE4cAE1uIE6NICBgid4gVfr0Xq23qz3WWvOms/swy9YH99R0ZHB ω = 15 ε / 8 AAAB9XicbVDJSgNBEK2JW4xb1KOXwSB4ijOuuQhBLx4jmAUyY+jp9CRNenqG7holDPkPLx4U8eq/ePNv7CwHjT4oeLxXRVW9IBFco+N8WbmFxaXllfxqYW19Y3OruL3T0HGqKKvTWMSqFRDNBJesjhwFayWKkSgQrBkMrsd+84EpzWN5h8OE+RHpSR5yStBI9x72GZJL98RL+FGlUyw5ZWcC+y9xZ6QEM9Q6xU+vG9M0YhKpIFq3XSdBPyMKORVsVPBSzRJCB6TH2oZKEjHtZ5OrR/aBUbp2GCtTEu2J+nMiI5HWwygwnRHBvp73xuJ/XjvFsOJnXCYpMkmni8JU2Bjb4wjsLleMohgaQqji5lab9okiFE1QBROCO//yX9I4Lrvn5bPb01L1ahZHHvZgHw7BhQuowg3UoA4UFDzBC7xaj9az9Wa9T1tz1mxmF37B+vgGTr+Rvw== ω = 13 ε / 8 (c) AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4Kon4dSx68diCaQttKJvtpF272YTdjVBKf4EXD4p49Sd589+4bXPQ1gcDj/dmmJkXpoJr47rfzsrq2vrGZmGruL2zu7dfOjhs6CRTDH2WiES1QqpRcIm+4UZgK1VI41BgMxzeTf3mEyrNE/lgRikGMe1LHnFGjZXqfrdUdivuDGSZeDkpQ45at/TV6SUsi1EaJqjWbc9NTTCmynAmcFLsZBpTyoa0j21LJY1RB+PZoRNyapUeiRJlSxoyU39PjGms9SgObWdMzUAvelPxP6+dmegmGHOZZgYlmy+KMkFMQqZfkx5XyIwYWUKZ4vZWwgZUUWZsNkUbgrf48jJpnFe8q8pl/aJcvc3jKMAxnMAZeHANVbiHGvjAAOEZXuHNeXRenHfnY9664uQzR/AHzucPtG2M4w== U AAAB9XicbVDLSsNAFL2pr1pfVZduBovgQkoiProsCuKyhaYttLFMppN26GQSZiZKCfkPNy4Uceu/uPNvnD4W2nrgwuGce7n3Hj/mTGnb/rZyK6tr6xv5zcLW9s7uXnH/oKmiRBLqkohHsu1jRTkT1NVMc9qOJcWhz2nLH91O/NYjlYpFoqHHMfVCPBAsYARrIz24vbQrw7R+18jOKlmvWLLL9hRomThzUoI5ar3iV7cfkSSkQhOOleo4dqy9FEvNCKdZoZsoGmMywgPaMVTgkCovnV6doROj9FEQSVNCo6n6eyLFoVLj0DedIdZDtehNxP+8TqKDipcyESeaCjJbFCQc6QhNIkB9JinRfGwIJpKZWxEZYomJNkEVTAjO4svLpHledq7Kl/WLUvVmHkcejuAYTsGBa6jCPdTABQISnuEV3qwn68V6tz5mrTlrPnMIf2B9/gARH5I+ U QFT , 8 AAAB9XicbVDLSsNAFL2pr1pfVZduBovgQkoi9bEsCuKyhaYttLFMppN26GQSZiZKCfkPNy4Uceu/uPNvnD4W2nrgwuGce7n3Hj/mTGnb/rZyK6tr6xv5zcLW9s7uXnH/oKmiRBLqkohHsu1jRTkT1NVMc9qOJcWhz2nLH91O/NYjlYpFoqHHMfVCPBAsYARrIz24vbQrw7R+18jOKlmvWLLL9hRomThzUoI5ar3iV7cfkSSkQhOOleo4dqy9FEvNCKdZoZsoGmMywgPaMVTgkCovnV6doROj9FEQSVNCo6n6eyLFoVLj0DedIdZDtehNxP+8TqKDay9lIk40FWS2KEg40hGaRID6TFKi+dgQTCQztyIyxBITbYIqmBCcxZeXSfO87FyWL+qVUvVmHkcejuAYTsGBK6jCPdTABQISnuEV3qwn68V6tz5mrTlrPnMIf2B9/gALC5I6 U QFT , 4 AAAB9HicbVBNS8NAEJ34WetX1aOXxSJ4kJJI/TgWPehFqGDaQhvKZrtpl242cXdTKCG/w4sHRbz6Y7z5b9y2OWjrg4HHezPMzPNjzpS27W9raXlldW29sFHc3Nre2S3t7TdUlEhCXRLxSLZ8rChngrqaaU5bsaQ49Dlt+sObid8cUalYJB71OKZeiPuCBYxgbSTP7aYdGaa399lpNeuWynbFngItEicnZchR75a+Or2IJCEVmnCsVNuxY+2lWGpGOM2KnUTRGJMh7tO2oQKHVHnp9OgMHRulh4JImhIaTdXfEykOlRqHvukMsR6oeW8i/ue1Ex1ceSkTcaKpILNFQcKRjtAkAdRjkhLNx4ZgIpm5FZEBlphok1PRhODMv7xIGmcV56Jy/lAt167zOApwCEdwAg5cQg3uoA4uEHiCZ3iFN2tkvVjv1sesdcnKZw7gD6zPH11Ckdk= U GM , 4 AAAB9HicbVBNS8NAEJ34WetX1aOXxSJ4kJKIHz0WPehFqGDaQhvKZrtpl242cXdTKCG/w4sHRbz6Y7z5b9y2OWjrg4HHezPMzPNjzpS27W9raXlldW29sFHc3Nre2S3t7TdUlEhCXRLxSLZ8rChngrqaaU5bsaQ49Dlt+sObid8cUalYJB71OKZeiPuCBYxgbSTP7aYdGaa399lpNeuWynbFngItEicnZchR75a+Or2IJCEVmnCsVNuxY+2lWGpGOM2KnUTRGJMh7tO2oQKHVHnp9OgMHRulh4JImhIaTdXfEykOlRqHvukMsR6oeW8i/ue1Ex1UvZSJONFUkNmiIOFIR2iSAOoxSYnmY0MwkczcisgAS0y0yaloQnDmX14kjbOKc1m5eDgv167zOApwCEdwAg5cQQ3uoA4uEHiCZ3iFN2tkvVjv1sesdcnKZw7gD6zPH2NWkd0= U GM , 8 AAAB9XicbVDLSsNAFL2pr1pfVZduBovgQkoi9bEsCuKyhaYttLFMppN26GQSZiZKCfkPNy4Uceu/uPNvnD4W2nrgwuGce7n3Hj/mTGnb/rZyK6tr6xv5zcLW9s7uXnH/oKmiRBLqkohHsu1jRTkT1NVMc9qOJcWhz2nLH91O/NYjlYpFoqHHMfVCPBAsYARrIz24vbQrw7R+18jOKlmvWLLL9hRomThzUoI5ar3iV7cfkSSkQhOOleo4dqy9FEvNCKdZoZsoGmMywgPaMVTgkCovnV6doROj9FEQSVNCo6n6eyLFoVLj0DedIdZDtehNxP+8TqKDay9lIk40FWS2KEg40hGaRID6TFKi+dgQTCQztyIyxBITbYIqmBCcxZeXSfO87FyWL+qVUvVmHkcejuAYTsGBK6jCPdTABQISnuEV3qwn68V6tz5mrTlrPnMIf2B9/gALC5I6 U QFT , 4 AAAB9HicbVBNS8NAEJ34WetX1aOXxSJ4kJJI/TgWPehFqGDaQhvKZrtpl242cXdTKCG/w4sHRbz6Y7z5b9y2OWjrg4HHezPMzPNjzpS27W9raXlldW29sFHc3Nre2S3t7TdUlEhCXRLxSLZ8rChngrqaaU5bsaQ49Dlt+sObid8cUalYJB71OKZeiPuCBYxgbSTP7aYdGaa399lpNeuWynbFngItEicnZchR75a+Or2IJCEVmnCsVNuxY+2lWGpGOM2KnUTRGJMh7tO2oQKHVHnp9OgMHRulh4JImhIaTdXfEykOlRqHvukMsR6oeW8i/ue1Ex1ceSkTcaKpILNFQcKRjtAkAdRjkhLNx4ZgIpm5FZEBlphok1PRhODMv7xIGmcV56Jy/lAt167zOApwCEdwAg5cQg3uoA4uEHiCZ3iFN2tkvVjv1sesdcnKZw7gD6zPH11Ckdk= U GM , 4 FIG. 1. (a) A linear-optical unitary U —either an 8-th order Quan tum F ourier T ransform (QFT) or an 8-mo de Hadamard ‘Green Mac hine’ (GM) circuit—receiv es in the ﬁrst input mo de the single-rail qubit that w e wish to measure in the |± ϕ ⟩ ≡ ( | 0 ⟩ ± e iϕ | 1 ⟩ ) / √ 2 basis. Remaining inputs are fed | + ϕ ⟩ ancillas that help b oost the success probability of measuring in the desired basis. (b) The GM circuit U GM is realized with a mesh of 50-50 b eam splitters and (c) the QFT U QFT adds phase shifters, following the construction from Ref. [ 31 ]. phases as in the QFT unitary of the corresp onding size. W e calculate the success probabilities for the ab o ve- men tioned approac h b y considering up to n = 10 QFTs and n = 8 Hadamard unitaries, and deduce a general form ula for the success rates for general n , with the con- clusion that the maximum success rate is ab out 57% and is obtained for the n = 8 QFT or Green Machine (see Fig. 1 ), thereby making this a near-term implemen table idea, given the recen t exp erimental realization of the n = 16 Green Machine for demonstrating sup er-additive capacit y [ 34 ]. Our ﬁndings can b e summarized as follows: 1. The p ow er-of-2 Hadamard unitaries yield the same success rate as the QFTs of the same size n . 2. F or even n , the success rate slowly increases from 50% (for n = 2) up to a maximum v alue of 147 / 256 ≈ 0 . 5742 for n = 8. On increasing n further, it starts decreasing again, asymptotically falling bac k to 50% in the large n limit for even n . 3. F or n = 4 and n = 6, w e get close to the ab o ve- men tioned maximum success rate, i.e., 9 / 16 = 0 . 5625 and 55 / 96 ≈ 0 . 5729, resp ectiv ely . 4. F or o dd n , the success rate is ( n − 1) / 2 n , whic h approac hes 50% from b elow for large n . 5. The measurement technique is asymmetric. F or n = 2, we hav e 50% success rate for unambigu- ously measuring b oth |±⟩ . But for higher n v alues, the success rate for measuring |−⟩ slowly rises and approac hes 100% for n → ∞ . But the probability of successfully measuring a qubit in | + ⟩ is 0 for odd n , and for ev en n , it slo wly decreases with n and asymptotically falls to 0 as n → ∞ . 3 6. if we use |−⟩ ancillas instead of | + ⟩ , then the success rates for | + ⟩ and |−⟩ are in terchanged. Therefore, if w e are measuring several copies of a system, we can conduct half of our measuremen ts with | + ⟩ ancillas and the rest with |−⟩ to obtain symmetric results. 7. W e pro ve that the abov e generalizes straightfor- w ardly to measuring in the |± ϕ ⟩ = ( | 0 ⟩ ± e iϕ | 1 ⟩ ) / √ 2 basis if w e emplo y | + ϕ ⟩ ancillary states instead of | + ⟩ ones, and the success rates are indep enden t of the v alue of ϕ . The simpler approac h for measur- ing in the |± ϕ ⟩ basis, how ev er, w ould b e to apply a e − iϕ optical phase on the single-rail qubit and measure it in the X basis using | + ⟩ ancillas. Calculating the suc c ess and failur e r ates. —Consider the set-up for measuring a single-rail qubit in the |± ϕ ⟩ basis b y feeding the optical mode enco ding this qubit into the ﬁrst input of an n -comp onent linear in terferometer, with the optical mo des enco ding | + ϕ ⟩ ancillary qubits in the remaining input p orts. Going forw ard, w e will at times refer to the single-rail qubit and the optical mode enco ding it interc hangeably , since the meaning will b e clear from con text. W e describ e ho w the | + ϕ ⟩ ancillas can b e generated in the Supplementary Material. The measurement in the |± ϕ ⟩ basis thus turns in to the problem of discriminating b etw een | + ϕ ⟩ ⊗ n and |− ϕ ⟩ ⊗ | + ϕ ⟩ ⊗ ( n − 1) b y feeding these in to a linear in terferome- ter and measuring the output photon click patterns. In terms of the creation op erators, these states can b e ex- pressed as | ± ϕ, 1 + ϕ, 2 + ϕ, 3 . . . + ϕ,n ⟩ = 1 2 n/ 2 (1 ± e iϕ a † 1 ) × n Y j =2 (1 + e iϕ a † j ) | 0 1 0 2 . . . 0 n ⟩ . (2) Here |± ϕ,j ⟩ denotes qubit n um b er j in the state |± ϕ ⟩ , and a † j the photon creation op erator for the optical mode enco ding the j -th single-rail qubit. A linear interferometer transforms the creation op- erators as a † j 7→ P n k =1 U j k b † k , where b † k are the cre- ation operators in the output, and U the unitary transformation matrix. The output state is then N ± ( ϕ, b † 1 , b † 2 , . . . , b † n ) | 0 1 0 2 . . . 0 n ⟩ , where N ± ( ϕ, b † 1 , b † 2 , . . . , b † n ) ≡ 1 2 n/ 2 1 ± e iϕ n X k 1 =1 U 1 k 1 b † k 1 ! × n Y j =2 1 + e iϕ n X k i =1 U j k i b † k j ! . (3) The op erators N ± ( ϕ, b † 1 , b † 2 , . . . , b † n ) are thus p olynomials in terms of the output creation op erators of the form N ± ( ϕ, b † 1 , b † 2 , . . . , b † n ) = n X i 1 ,...i n =0 , P n k =1 i k ≤ n e P n k =1 i k ϕ c ± ,i 1 i 2 ...i n n Y j =1 ( b † j ) i j . (4) The probability of obtaining the clic k pattern | i 1 i 2 i 3 . . . i n ⟩ on measuring the interferometer outputs is P ± ,i 1 i 2 ...i n = | c ± ,i 1 i 2 ...i n | 2 i 1 ! i 2 ! . . . i n ! , (5) where the factor of i 1 ! i 2 ! . . . i n ! arises from the fact that ( b † j ) i j | 0 j ⟩ = p i j ! | i j ⟩ . Th us, these probabilities are clearly indep endent of the v alue of ϕ , and hence w e will obtain the same success and failure rates for measuring a single-rail qubit in |± ϕ ⟩ regardless of the v alue of ϕ . F rom a practical experimental persp ective, it mak es more sense to apply a single e − iϕ phase gate on our single-rail qubit to rotate the measurement basis from |± ϕ ⟩ to ±⟩ , and measure in the latter by employing a collection of | + ⟩ ancillary states. In this w ay , we only need to apply one phase gate on the original single-rail qubit, rather than in tro ducing a e iϕ phase in each of the ancillary qubits. Note that as n increases, ﬁnding the coeﬃcients be- comes a very lab orious exercise with the n umber of terms in the p olynomials N ± ( ϕ, b † 1 , b † 2 , . . . , b † n ) scaling exp onen- tially , reminiscent of Boson Sampling [ 35 ]. One relativ ely simple (though not necessarily fast) wa y to obtain the probabilities is to use the symbolic manipulation fea- tures of a mathematical soft ware suc h as Maple , Mat- lab or Mathematic a to ﬁnd the coeﬃcients c ± ,i 1 i 2 ...i n as the T aylor series coeﬃcients of the p olynomials ( 4 ). W e can then calculate the probabilities of the v arious clic k patterns according to ( 5 ). W e need to run an iterative routine that go es through all the clic k patterns individu- ally , and computes their probabilities. If P ± ,i 1 i 2 ...i n = 0 and P ∓ ,i 1 i 2 ...i n  = 0, then the click pattern | i 1 , i 2 , . . . i n ⟩ means our single-rail qubit was unambiguously measured and found to b e in |∓ ϕ ⟩ in the idealized, noiseless and lossless case. If both P ± ,i 1 i 2 ...i n  = 0, then the click pat- tern represents failure of measurement in the |± ϕ ⟩ basis. W e need to add up all the success and failure probabili- ties asso ciated with the diﬀerent output click patterns to obtain the ov erall success rate for measuring our qubit in the |± ϕ ⟩ basis. W e describ e our algorithm for this purp ose in the supplemen tary material [ 36 ]. As w e shall discuss shortly , w e ﬁnd that the success rates are the same for QFTs and the corresp onding pow er of 2 Hadamard unitaries. Therefore, w e denote the suc- cess and failure rates for measuring a qubit in |± ϕ ⟩ b y s n ± and f n ± , resp ectiv ely , for b oth QFTs and Hadamard co des, and indep enden t of the angle ϕ since we hav e sho wn that the probabilities of the output click patterns are indep enden t of ϕ . Additionally , to gain further in- sigh t into the behavior of the success and failure rates, w e can classify the v arious output states | i 1 i 2 . . . i n ⟩ in terms of the total photon count I = i 1 + i 2 + . . . + i n in 4 the output click patterns. Since a lossless and noiseless linear in terferometer conserves the total photon num b er, w e hav e the same total probability P n, I of obtaining one of the click patterns with I photons as the probabilit y of I photons in the input state. W e can then consider the conditional success and failure rates of our |± ϕ ⟩ basis measuremen t given that we obtain a click pattern with I photons for more insight in to the b ehavior of the results as well as apply v arious consistency chec ks. W e describ e this in more detail in the supplementary material. Using |−⟩ ancil las inste ad of | + ⟩ ones. —Note in Eq. ( 2 ) that if we insert a minus sign in front of each input photon creation op erator, i.e. tak e a † j → − a † j , the input states change from | ± ϕ, 1 + ϕ, 2 + ϕ, 3 . . . + ϕ,n ⟩ to | ∓ ϕ, 1 − ϕ, 2 − ϕ, 3 . . . − ϕ,n ⟩ , corresp onding to the use of |−⟩ ancillas with the ﬁrst qubit ﬂipp ed. This only changes the output state by resulting in all the output p ort pho- ton creation op erators acquiring minus signs. The terms in N ± ( ϕ, b † 1 , b † 2 , . . . , b † n ) corresp onding to ev en v alues of the total photon count I = i 1 + i 2 + . . . i n , remain ex- actly unchanged, whereas those with o dd v alues of I ac- quire physically inconsequen tial minus signs. Thus, the probabilities P + ,i 1 i 2 ...i n and P − ,i 1 i 2 ...i n asso ciated with our single-rail qubit b eing in | + ⟩ and |−⟩ , resp ectively , are switched if we use |−⟩ ancillas instead of | + ⟩ ones. Th us the success rates for measuring | + ⟩ and |−⟩ are also switched. W e hav e also numerically found b y trying several ex- amples that randomly switc hing only some of the ancillas from | + ϕ ⟩ to |− ϕ ⟩ while keeping the rest in | + ϕ ⟩ often results in a reduction in the ov erall success rate. This is not particularly surprising, since having all the ancil- las in either | + ϕ ⟩ or |− ϕ ⟩ creates symmetries which are helpful for scrambling our single-rail qubit in a wa y that impro ves the success rate. Randomly ha ving some qubits in | + ϕ ⟩ and others in |− ϕ ⟩ does not app ear to maintain the same scram bling. How ev er, a full study considering all such cases is practically imp ossible, and w e do not explore this topic further. QFTs and Hadamar d unitaries. —The unitary trans- formation matrix for an n -comp onent QFT has the en- tries [ 27 ] ( U QFT , n , ) j,k ≡ ω ( j − 1)( k − 1) / √ n , where ω = exp(2 iπ /n ). The Hadamard matrices, on the other hand, are deﬁned as n × n matrices comprising en tirely of ‘ones’ and ‘minus ones’, and satisfy the conditions H n H ⊤ n = nI n and det( H n ) = ± √ n , where I n is the n × n identit y matrix. Dividing H n b y √ n therefore gives a unitary transformation. When n is a p ow er of 2, the Hadamard unitary matrix is deﬁned recursively in terms of the n 2 comp onen t Hadamard unitary [ 32 ] U GM , n = 1 √ 2  U GM , n / 2 U GM , n / 2 U GM , n / 2 − U GM , n  , (6) for n = 2 j , j ∈ { 1 , 2 , . . . } , where U GM , 2 = 1 √ 2  1 1 1 − 1  , (7) and is implemen ted by a balanced beam splitter. In the supplemen tary material accompanying this work, we de- scrib e ho w larger Hadamard unitaries can be constructed from suc h beam splitters [ 6 ]. W e should men tion that Hadamard matrices ha ve also been shown to exist for sev- eral num b ers other than pow ers of 2 with the general con- jecture that these exist for all m ultiples of 4 [ 33 ]. How- ev er, these do not ha ve a nice recursiv e form as ab ov e, and w e only consider the n = 12 case of this brieﬂy in this w ork with the ﬁnding that it only giv es a low success rate of ab out 36 . 63% of measuring a single-rail qubit in the |± ϕ ⟩ basis. R esults. —W e hav e calculated the success and fail- ure rates for QFTs up to n = 10 and the p ow er-of-2 Hadamard unitaries up to n = 8. These results hav e the follo wing patterns (shown in Fig. 2 ): 1. F or even n , the success rate for measuring | + ϕ ⟩ is s n + = n ! 2 n [( n/ 2)!] 2 , (8) whic h decreases with n and go es to 0 as n → ∞ . F or o dd n , the success rate for measuring | + ϕ ⟩ is exactly zero for all n . 2. The success rate for measuring |− ϕ ⟩ for all (ev en or o dd) n is s n − = n − 1 n , (9) whic h approaches unity as n → ∞ . 3. F or even n , the o v erall success rate (whic h is the a verage of success rates for | + ϕ ⟩ and |− ϕ ⟩ if we assume equal prior probabilities for both,) is then giv en by s n, ov erall = 1 2  n ! 2 n [( n/ 2)!] 2 + n − 1 n  for ev en n (10) This is equal to 0 . 5 for n = 2, 9 / 16 = 0 . 5625 for n = 4, 55 / 96 ≈ 0 . 5729 for n = 6, and reaches a maximum of 147 / 256 ≈ 0 . 5742 for n = 8. Af- ter that, it slowly decreasing and asymptotically approac hes 50% from ab ov e as n → ∞ instead of increasing and approac hing unity as speculated b y [ 14 ]. F or o dd n , the ov erall success rate (again, assuming equal priors,) is s n, ov erall = n − 1 2 n for o dd n (11) This asymptotically approaches 50% from below as n → ∞ . 5 5 10 15 20 25 n 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 s n s n + s n ° s n, ov erall s n + s n ° s n, o v erall s n + s n ° s n, o v erall s n + s n ° s n, ov erall even n odd n FIG. 2. Success probabilities s n of measuring |± ϕ ⟩ as well as the ov erall success, as a function of the num ber of inputs n of the Green Machine (only applies for n = 2 j , j ∈ { 1 , 2 , 3 , . . . } ) and QFT (all v alues of n apply). The maximum ov erall suc- cess is found to b e s 8 , ov erall = 147 / 256 or ∼ 57 . 42%. Some further observ ations based on the break-do wn in terms of the total n umber of photons in the v arious out- put click patterns are given in the supplementary mate- rial [ 36 ]. The fact that the success and failure rates are asym- metric b et w een | + ϕ ⟩ and |− ϕ ⟩ is not en tirely surprising giv en that the ancillas are all | + ϕ ⟩ . Recall our earlier discussion that if w e use |− ϕ ⟩ for all the ancillas, then the success rates for | + ϕ ⟩ and |− ϕ ⟩ are swapped. There- fore, in a setting where w e measure multiple copies of a single-rail photon in the |± ϕ ⟩ basis, it may b e worth carrying out half of the measuremen ts with | + ϕ ⟩ ancillas, and the other half with |− ϕ ⟩ ancillas. How ever, as men- tioned earlier, we hav e also brieﬂy considered using | + ϕ ⟩ for some of the ancillas, and and |− ϕ ⟩ for the rest, and ha ve found that it does not lead to any impro v ement and in some cases ev en results in reducing the success rates, though we do not hav e a rigorous proof and cannot rule out p ossible exceptions with 100% conﬁdence. Conclusion. —W e ha ve inv estigated the problem of measuring a single-rail qubit in the |± ϕ ⟩ basis along an arbitrary axis of the X Y plane of the qubit Blo ch sphere b y employing n − 1 unentangled single-rail ancillas in the | + ϕ ⟩ state, along with an n -component QFT or the p o w er-of-2 Green Machine realizing the Hadamard uni- tary . Our prop osed prescription is to feed the optical mo de enco ding the single-rail qubit into the ﬁrst input p ort of a QFT (or a p o wer-of-2 Green Machine), with single-rail | + ϕ ⟩ ancilla states fed in to the other inputs, and detect photons in all outputs using photon num ber resolving detectors. By calculating the success rates for up to n = 10, w e observe the patterns and deduce a conjecture for the general formula for the success rate for arbitrary n . W e obtain the same success rate for an n -component Green Machine and the QFT when n is a p o wer of 2, and pro ve that it is independent of the v alue of the azimuthal angle ϕ . W e ﬁnd that this strat- egy b oosts the measurement success rate up to a maxi- m um of 147 / 256 ≈ 57 . 42% for n = 8, but do es not give a fully deterministic measuremen t that Ref. [ 14 ] conjec- tured. Moreo ver, the simpler n = 4 Green Machine or QFT comes fairly close to this maxim um v alue with a suc- cess rate of 9 / 16 = 56 . 25%, and is readily realizable [ 34 ]. The measurement strategy is, how ev er, very asymmet- ric. Except the 50% success probability for n = 2, the probabilities for successfully measuring a qubit that is in | + ϕ ⟩ and |− ϕ ⟩ are uneven. The success rates for the latter are n/ ( n − 1) for all (even and odd) n . But those for the former slo wly fall to zero for large n if n is ev en, and are exactly 0 for all o dd v alues of n . Thus in the large n limit, the probabilities for successfully measur- ing | + ϕ ⟩ and |− ϕ ⟩ approach zero and 100%, resp ectively . Therefore, if we ha ve an application where we care more ab out measuring |− ϕ ⟩ , then this can still be a fairly useful strategy . On the other hand, for more general applica- tions where we need to measure b oth | + ϕ ⟩ and |− ϕ ⟩ , this metho d will b e inadequate due to its falling success rate for | + ϕ ⟩ . That said, we ha v e also sho wn that if we use |−⟩ ancillas instead of | + ⟩ ones, then the success rates of measuring | + ϕ ⟩ and |− ϕ ⟩ are simply sw apped. Therefore, w e can still b eneﬁt from the gain ab o ve 50% for n > 2 in settings where we are measuring m ultiple copies of the same quantum system by emplo ying | + ϕ ⟩ ancillas for half of the trials and |− ϕ ⟩ for the rest. The fact that we do not obtain a 100% success rate, ho wev er, lea ves op en how to carry out a deterministic single-rail measurement along some direction on the X Y plane of the Blo c h sphere such as the X or Y bases. Two solutions that attain 100% success rates in the idealized noiseless case will b e presen ted in a separate work [ 37 ], but these inv olv e entangled ancillas. Therefore, in the presence of resource constraints, the simplicit y of a 50-50 b eam splitter or an n = 4 Green Mac hine with | + ϕ ⟩ or |− ϕ ⟩ ancillas remains an attractive practical option. It would b e remiss not to mention a long line of liter- ature on Bell-basis measurements on dual-rail photonic qubits, i.e., a qubit enco ded b y the presence of a single photon in one of tw o orthogonal modes: a no-go theo- rem that linear-optics alone cannot surpass a 50% success rate [ 38 ], bo osting the success rate to up to 25 / 32 using unen tangled single-photon ancillas [ 39 ] and to 59 . 6% us- ing in-line squeezing [ 40 , 41 ], and recen t extensions to log- ical Bell measuremen ts [ 42 ] and exp eriments [ 43 ]. Beat- ing this 50% limit has had profound implications to the resource eﬃciency of photonic cluster state generation via p ercolation theory [ 44 , 45 ]. Given the close relationship b et w een b o osting partial Bell-state analysers and single- qubit measuremen ts as discussed ab o v e, and the close synergies of the ev olution of m ulti-photon en tanglemen t 6 in linear interferometers, one may be able to borrow ideas from this literature into the single-rail qubit domain. The authors ar e truly gr ateful to Johannes Borr e- gaar d for se e ding the original ide a during a discus- sion with SG at The Hague, and for suggesting the use of c oher ent-state ancil las. The authors thank Joseph Gabriel Richar dson for many helpful c onversa- tions thr oughout the c ourse of this work. AS and SG acknow le dge funding supp ort fr om the D ARP A Phenom pr oje ct awar de d under c ontr act# HR00112490451, and the AR O Quantum Network Scienc e MURI awar de d un- der gr ant# W911NF2110325. IP and SG acknow le dge funding supp ort fr om AFOSR gr ant# F A9550-22-1-0180. ∗ asa jjad@umd.edu † iacpad0795@arizona.edu ‡ saik at@umd.edu [1] V. V. 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Guha, Supple- men tal Material for “Bo osted linear-optic measurements on single-rail qubits with unentangled ancillas”, Av ail- able as Supplemental Material ﬁle accompanying this man uscript (2025), includes additional deriv ations, ﬁg- ures, and discussion. [37] A. Sa jjad and S. Guha, Unpublished (2025). [38] J. Calsamiglia and N. L ¨ utk enhaus, Appl. Phys. B 72 , 67 (2001). [39] F. Ewert and P . v an Lo ock, Ph ys. Rev. Lett. 113 , 140403 (2014). [40] H. A. Zaidi and P . v an Lo ock, Ph ys. Rev. Lett. 110 , 260501 (2013). [41] T. Kilmer and S. Guha, Phys. Rev. A 99 , 032302 (2019). [42] P . Hilaire, Y. Castor, E. Barnes, S. E. Economou, and F. Grosshans, PRX quantum 4 , 040322 (2023). [43] N. Hauser, M. J. Bay erbac h, S. E. D’Aurelio, R. W eb er, M. Santandrea, S. P . Kumar, I. Dhand, and S. Barz, Np j Quan tum Inf. 11 , 1 (2025). [44] M. Gimeno-Segovia, P . Shadb olt, D. E. Bro wne, and T. Rudolph, Phys. Rev. Lett. 115 , 020502 (2015). [45] M. Pan t, D. T owsley , D. Englund, and S. Guha, Nat. Comm un. 10 , 1070 (2019). [46] R. V asconcelos, S. Reisen bauer, C. Salter, G. W ach ter, 7 D. Wirtitsch, J. Schmiedma yer, P . W alther, and M. T rupk e, np j Quantum Information 6 , 9 (2020) . [47] The idea of employing a coheren t state ancilla and mix- ing it with the optical mo de enco ding a single-rail qubit, w as describ ed in [ 14 ? ] in the con text of loading the join t state of a star light photon arriving across tw o dis- tan t telescop es on to quantum memories at the t wo sites. In that set up, the prop osal was to emplo y a coherent state | α ⟩ at each site, mix it with the optical mo de of the star light photon at that lo cation using a balanced b eam splitter, and measure the outputs with photon num b er detectors. Then Alice and Bob share their measurement results with each other through classical communication to determine if the tw o-telescop e state of the astronom- ical photon has b een correctly loaded on to quantum memories at the tw o sites, loaded with a min us sign that needs to b e corrected with a Z gate, or if there is a fail- ure in loading. Here, we consider the problem of making an X basis measurement on just one single-rail qubit, or loading its state on to a quantum memory b y emplo ying a coheren t state ancilla as opposed to the abov e-men tioned t wo-site scenario. 8 SUPPLEMENT AR Y MA TERIAL Motiv ation: the state transfer protocol and X basis measuremen t Consider a single-rail qubit in some arbitrary state υ | 0 p ⟩ + ξ | 1 p ⟩ , which we wish to transfer to an atomic memory . F ollowing the circuit telep ortation proto col [ 27 ], we can bring in an atomic qubit initialized in | 0 a ⟩ . A CNOT gate from the single-rail qubit to the atomic one entangles the t w o qubits, giving the join t state υ | 0 p 0 a ⟩ + ξ | 1 p 1 a ⟩ . It is straigh tforward to see that an X basis measurement of the photonic qubit leav es the atomic qubit in υ | 0 a ⟩ ± ξ | 1 a ⟩ , corresp onding to the outcome |± p ⟩ . Thus, the state transfer is complete if the measurement outcome is | + p ⟩ , whereas w e need to apply a corrective Z gate if w e obtain |− p ⟩ . In theory , this proto col can b e implemen ted using the Duan-Kim ble metho d [ 28 ], where the atomic qubit is deﬁned in terms of the spin states of an atom trapp ed inside a ca vity , and the optical mo de enco ding the single-rail qubit is allow ed to interact with this atom. If there is a photon, it reﬂects oﬀ of the | ↓⟩ spin state, imparting a π phase to it, while lea ving the | ↑⟩ state inv ariant. Deﬁning the states ( | ↑⟩ + | ↓⟩ ) / √ 2 and ( | ↑⟩ − | ↓⟩ ) / √ 2 to b e the logical | 0 a ⟩ and | 1 a ⟩ states of the atomic qubit, resp ectiv ely , this arrangemen t giv es us a CNOT gate from the single-rail qubit to the atomic one. An experimental demonstration of suc h an interaction of an optical mo de with an atom in a cavit y has b een presented in [ 29 ] in the context of dual-rail qubits, but realized with weak-coheren t state appro ximate time-bin dual-rail qubits. Generating single-rail | + ⟩ or | + ϕ ⟩ ancilla qubits W e can generate a single-rail | + ⟩ or more generally , a | + ϕ ⟩ state with color centers such as Nitrogen V acancy (NV) or Silicon V acancy (SiV) cen ters, by using a v ariant of the protocol describ ed in [ 46 ] to produce dual-rail | + ⟩ states. A color center eﬀectively b eha ves as a tw o-lev el atom in whic h the ground state splits into the spin up | ↑ s ⟩ and spin do wn | ↓ s ⟩ states in the presence of a static magnetic ﬁeld. The excited state also splits in to its spin-up and spin-down v ersions | ↑ ′ ⟩ and | ↓ ′ ⟩ , and the energy gap b etw een them is diﬀerent from that b et ween the t wo spin states of the ground state. The system is initialized in a tensor pro duct of the ‘spin down’ atomic state and the v acuum photonic state | ↓ s ⟩| 0 p ⟩ , follo wed by a π / 2 microw a ve pulse on the former to ha ve the superp osition | γ sp ⟩ = 1 √ 2 ( | ↑ s ⟩| 0 p ⟩ + | ↓ s ⟩| 0 p ⟩ ) . (12) W e can then apply a pulse resonant with the transition | ↓⟩ → | ↓ ′ ⟩ . As the atom falls bac k to the ground state | ↓ s ⟩ , a photon is emitted, resulting in the spin-photon entangled state | γ ′ sp ⟩ = 1 √ 2 ( | ↑ s ⟩| 0 p ⟩ + | ↓ s ⟩| 1 p ⟩ ) = 1 √ 2 ( | + s ⟩| + p ⟩ + |− s ⟩|− p ⟩ ) (13) where |± s ⟩ ≡ ( | ↑ s ⟩ ± | ↓ s ⟩ ) / √ 2 are the X basis states of the spin qubit. Measuring out the atomic qubit in the X basis th us gives us a single-rail | + ⟩ state if the outcome is | + s ⟩ . If the outcome is |− s ⟩ , then we obtain a single-rail |−⟩ state, which we can ﬂip into a | + ⟩ by applying a Z op eration. In the same spirit, we can apply a ± e iϕ phase dela y on a |±⟩ single-rail qubit to turn it into a | + ϕ ⟩ state. Al- ternativ ely , we can apply a phase shift gate with the phase e iϕ on the atomic qubit in | γ ′ sp ⟩ , so that | γ ′ sp ⟩ turns in to | γ ′ sp,ϕ ⟩ = 1 √ 2 ( | ↑ s ⟩| 0 p ⟩ + e iϕ | ↓ s ⟩| 1 p ⟩ ) . (14) Rewriting this state in the spin X basis, we obtain | γ ′ sp,ϕ ⟩ = 1 √ 2 ( | + s ⟩| + p,ϕ ⟩ + |− s ⟩|− p,ϕ ⟩ ) . (15) where |± p,ϕ ⟩ deis the photonic |± ϕ ⟩ state. Measuring out the spin system in the X basis leav es the photonic qubit in the | + ϕ ⟩ state if the result is | + s ⟩ and |− ϕ ⟩ if the result is |− s ⟩ . 9 50:50 BS Green Machi ne for AAAB73icbVBNSwMxEJ31s9avqkcvwSJ4kLIrfh2LHvQiVHDbQruUbJptQ5PsmmSFsvRPePGgiFf/jjf/jWm7B219MPB4b4aZeWHCmTau++0sLC4tr6wW1orrG5tb26Wd3bqOU0WoT2Ieq2aINeVMUt8ww2kzURSLkNNGOLge+40nqjSL5YMZJjQQuCdZxAg2Vmr6nezm7liOOqWyW3EnQPPEy0kZctQ6pa92NyapoNIQjrVueW5iggwrwwino2I71TTBZIB7tGWpxILqIJvcO0KHVumiKFa2pEET9fdEhoXWQxHaToFNX896Y/E/r5Wa6DLImExSQyWZLopSjkyMxs+jLlOUGD60BBPF7K2I9LHCxNiIijYEb/bleVI/qXjnlbP703L1Ko+jAPtwAEfgwQVU4RZq4AMBDs/wCm/Oo/PivDsf09YFJ5/Zgz9wPn8Akm+Prg== U GM ,n AAAB73icbVBNS8NAEJ3Ur1q/qh69LBbBg5Sk+HUsetCLUMG0hTaUzXbTLt1s4u5GKCF/wosHRbz6d7z5b9y2OWjrg4HHezPMzPNjzpS27W+rsLS8srpWXC9tbG5t75R395oqSiShLol4JNs+VpQzQV3NNKftWFIc+py2/NH1xG89UalYJB70OKZeiAeCBYxgbaS220tv7k5qWa9csav2FGiRODmpQI5Gr/zV7UckCanQhGOlOo4day/FUjPCaVbqJorGmIzwgHYMFTikykun92boyCh9FETSlNBoqv6eSHGo1Dj0TWeI9VDNexPxP6+T6ODSS5mIE00FmS0KEo50hCbPoz6TlGg+NgQTycytiAyxxESbiEomBGf+5UXSrFWd8+rZ/WmlfpXHUYQDOIRjcOAC6nALDXCBAIdneIU369F6sd6tj1lrwcpn9uEPrM8fN0OPcg== U GM , 2 AAAB73icbVBNS8NAEJ3Ur1q/qh69LBbBg5Sk+HUsetCLUMG0hTaUzXbTLt1s4u5GKCF/wosHRbz6d7z5b9y2OWjrg4HHezPMzPNjzpS27W+rsLS8srpWXC9tbG5t75R395oqSiShLol4JNs+VpQzQV3NNKftWFIc+py2/NH1xG89UalYJB70OKZeiAeCBYxgbaS220tv7k5qWa9csav2FGiRODmpQI5Gr/zV7UckCanQhGOlOo4day/FUjPCaVbqJorGmIzwgHYMFTikykun92boyCh9FETSlNBoqv6eSHGo1Dj0TWeI9VDNexPxP6+T6ODSS5mIE00FmS0KEo50hCbPoz6TlGg+NgQTycytiAyxxESbiEomBGf+5UXSrFWd8+rZ/WmlfpXHUYQDOIRjcOAC6nALDXCBAIdneIU369F6sd6tj1lrwcpn9uEPrM8fN0OPcg== U GM , 2 AAAB73icbVBNS8NAEJ3Ur1q/qh69LBbBg5Sk+HUsetCLUMG0hTaUzXbTLt1s4u5GKCF/wosHRbz6d7z5b9y2OWjrg4HHezPMzPNjzpS27W+rsLS8srpWXC9tbG5t75R395oqSiShLol4JNs+VpQzQV3NNKftWFIc+py2/NH1xG89UalYJB70OKZeiAeCBYxgbaS220tv7k5qWa9csav2FGiRODmpQI5Gr/zV7UckCanQhGOlOo4day/FUjPCaVbqJorGmIzwgHYMFTikykun92boyCh9FETSlNBoqv6eSHGo1Dj0TWeI9VDNexPxP6+T6ODSS5mIE00FmS0KEo50hCbPoz6TlGg+NgQTycytiAyxxESbiEomBGf+5UXSrFWd8+rZ/WmlfpXHUYQDOIRjcOAC6nALDXCBAIdneIU369F6sd6tj1lrwcpn9uEPrM8fN0OPcg== U GM , 2 AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBg9Sk1I9j0YNehAqmLbahbLabdulmE3Y3Qgn9F148KOLVf+PNf+O2zUFbHww83pthZp4fc6a0bX9buaXlldW1/HphY3Nre6e4u9dQUSIJdUnEI9nysaKcCepqpjltxZLi0Oe06Q+vJ37ziUrFIvGgRzH1QtwXLGAEayM9ut305u5EnFbG3WLJLttToEXiZKQEGerd4lenF5EkpEITjpVqO3asvRRLzQin40InUTTGZIj7tG2owCFVXjq9eIyOjNJDQSRNCY2m6u+JFIdKjULfdIZYD9S8NxH/89qJDi69lIk40VSQ2aIg4UhHaPI+6jFJieYjQzCRzNyKyABLTLQJqWBCcOZfXiSNStk5L5/dV0u1qyyOPBzAIRyDAxdQg1uogwsEBDzDK7xZynqx3q2PWWvOymb24Q+szx90spAj U GM ,n/ 2 AAAB73icbVBNS8NAEJ3Ur1q/qh69LBbBg5Sk+HUsetCLUMG0hTaUzXbTLt1s4u5GKCF/wosHRbz6d7z5b9y2OWjrg4HHezPMzPNjzpS27W+rsLS8srpWXC9tbG5t75R395oqSiShLol4JNs+VpQzQV3NNKftWFIc+py2/NH1xG89UalYJB70OKZeiAeCBYxgbaS220tv7k5qWa9csav2FGiRODmpQI5Gr/zV7UckCanQhGOlOo4day/FUjPCaVbqJorGmIzwgHYMFTikykun92boyCh9FETSlNBoqv6eSHGo1Dj0TWeI9VDNexPxP6+T6ODSS5mIE00FmS0KEo50hCbPoz6TlGg+NgQTycytiAyxxESbiEomBGf+5UXSrFWd8+rZ/WmlfpXHUYQDOIRjcOAC6nALDXCBAIdneIU369F6sd6tj1lrwcpn9uEPrM8fN0OPcg== U GM , 2 AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBg9Sk1I9j0YNehAqmLbahbLabdulmE3Y3Qgn9F148KOLVf+PNf+O2zUFbHww83pthZp4fc6a0bX9buaXlldW1/HphY3Nre6e4u9dQUSIJdUnEI9nysaKcCepqpjltxZLi0Oe06Q+vJ37ziUrFIvGgRzH1QtwXLGAEayM9ut305u5EnFbG3WLJLttToEXiZKQEGerd4lenF5EkpEITjpVqO3asvRRLzQin40InUTTGZIj7tG2owCFVXjq9eIyOjNJDQSRNCY2m6u+JFIdKjULfdIZYD9S8NxH/89qJDi69lIk40VSQ2aIg4UhHaPI+6jFJieYjQzCRzNyKyABLTLQJqWBCcOZfXiSNStk5L5/dV0u1qyyOPBzAIRyDAxdQg1uogwsEBDzDK7xZynqx3q2PWWvOymb24Q+szx90spAj U GM ,n/ 2 . . . = AAACBHicbVC7TsMwFHXKq5RXgLGLRYXEEFVJeS5IFSyMRaIPqQmV4zqtW8eJbAepijqw8CssDCDEykew8Te4bQZoOZKlo3Pu0fU9fsyoVLb9beSWlldW1/LrhY3Nre0dc3evIaNEYFLHEYtEy0eSMMpJXVHFSCsWBIU+I01/eD3xmw9ESBrxOzWKiReiHqcBxUhpqWMW+WXlfmC51sCl3E0dq2IdW243UtIdd8ySXbangIvEyUgJZKh1zC+dxElIuMIMSdl27Fh5KRKKYkbGBTeRJEZ4iHqkrSlHIZFeOj1iDA+10oVBJPTjCk7V34kUhVKOQl9Phkj15bw3Ef/z2okKLryU8jhRhOPZoiBhUEVw0gjsUkGwYiNNEBZU/xXiPhIIK91bQZfgzJ+8SBqVsnNWPr09KVWvsjryoAgOwBFwwDmoghtQA3WAwSN4Bq/gzXgyXox342M2mjOyzD74A+PzB0IUlpk= n =2 j ,j → { 1 , 2 , 3 ,... } . . . . . . . . . . . . . . . . . . . . . . . . . . . FIG. 3. The Green Machine transformation for size n b eing p o wers-of-t w o can b e recursively constructed from Green Machines of size n = 2 (the balanced b eam splitter) and size n/ 2. Implemen ting pow er of 2 Hadamard unitaries Note that the 2-comp onen t Hadamard unitary U GM , 2 is implemented by a b eam splitter with the transformation a † 1 7→ 1 √ 2 ( b † 1 + b † 2 ) , a † 2 7→ 1 √ 2 ( b † 1 − b † 2 ) (16) b et w een the input creation operators a † 1 a † 2 and the output creation op erators b † 1 and b † 2 . An n -comp onent Green Mac hine can b e built recursively with such 50-50 b eam splitters (see Fig. 3 ) as follows: 1. Apply an n/ 2-comp onen t Green Mac hine on the ﬁrst n/ 2 p orts, and another n/ 2 comp onent Green Machine on the last n/ 2 p orts. 2. Apply a 50-50 b eam splitter of the form ( 16 ) to mix p ort 1 with port n/ 2 + 1. Another such b eam splitter to mix port 2 with n/ 2 + 2, and so on. Thus for eac h j from 1 to n/ 2, w e mix port j with n/ 2 + j with a 50-50 b eam splitter. It is w orth sp ecifying that this recursive construction only works when the b eam splitters are of the form ( 16 ). It will not work if we hav e the imaginary num b er i in the 2 × 2 transformation. Breaking down the success and failure rates in terms of the photon counts of the click patterns Before we describ e our algorithm for calculating the success and failure rates, we describ e how we will break down the analysis in terms of the total photon count I = i 1 + i 2 + . . . + i n for the output photon click patterns | i 1 i 2 . . . i n ⟩ . Recall that we are denoting the success and failure rates for measuring a qubit in |± ϕ ⟩ by s n ± and f n ± , resp ectively . Naturally , the sum of the probabilities b eing unity is a nice consistency chec k on our results: n X i 1 ,i 2 ...i n =1 P ± ,i 1 i 2 ...i n = 1 and s n ± + f n ± = 1 . (17) Additionally , w e can gain further insight into the behavior of the success and failure rates by classifying the v arious output states | i 1 i 2 . . . i n ⟩ in terms of the total photon count I = i 1 + i 2 + . . . + i n in the click patterns. Since the total num b er of photons is conserved in a lossless and noiseless linear interferometer, the total probability of output states with I total photons will b e the same as the probabilit y of I photons in the input states |± ϕ ⟩ ⊗ | + ϕ ⟩ ⊗ ( n − 1) . Sp eciﬁcally , if we write out each qubit from these input states in the Z basis, we obtain a total of 2 n terms, of which  n I  = n ! / [ I !( n − I )!] will hav e a total of I photons going in. These will hav e the total probability  n I  / 2 n , which will 10 b e equal to the total probability of output states with a total of I photons: P n, I ≡ 1 2 n  n I  = n ! I !( n − I )!2 n = n X i 1 ,i 2 ,...,i n =1 , i 1 + i 2 + ... + i n = I P ± ,i 1 i 2 ...i n (18) No w, let s n, I ± b e the probability of success of measuring and ﬁnding our input qubit 1 to b e in |± ϕ ⟩ with output clic k patterns that ha v e a total of I photons in all output p orts, and f n, I ± b e the corresp onding failure probability . Then P n, I = s n, I ± + f n, I ± , (19) and s n ± = n X I =0 s n, I ± , f n ± = n X I =0 f n, I ± . (20) This wa y , ( 19 ) and ( 20 ) provide additional consistency chec ks on our calculation of the probabilities of the v arious output click patterns as w ell as the success and failure rates for measuring our single rail qubit in | + ⟩ and |−⟩ . But more imp ortan tly , we will be analyzing the break-up of the success and failure rates in terms of the total photon coun ts in the output click patterns to deduce the general formulae for the success and failure rates. Our algorithm for calculating the success and failure rates First, recall that the probabilities for the output click patterns can be obtained b y writing down the p olynomials ( 4 ) in terms of the output port creation op erators and then obtaining the coeﬃcients c ± ,i 1 i 2 ...i n from a T a ylor series command in a computer algebra softw are like Maple , Mathematic a or Matlab . The probabilities for the individual clic k patterns are then given in terms of the co eﬃcients by ( 5 ). Since these probabilities are indep enden t of the v alue of ϕ , we will work with the ϕ = 0 case to simplify the calculation and av oid ha ving to deal with complex phases. T o determine the success rates from the probabilities, we can use the following pro cedure (summarized in algo- rithm 1 ): 1. T o add up the total success and failure probabilities, we introduce v ariables s n, I ± and f n, I ± and initialize them with v alues of 0 b efore starting an y of the lo ops. 2. W e already know that the 0 and n -photon click patterns give zero success rate. Hence, f n, 0 ± = P n, 0 f n,n ± = P n,n , (21) whic h we just assign in a single line instead of going through these click patterns. 3. Deﬁne an arra y of v ariables [ i 1 , i 2 , . . . , i n ]. 4. Run a set of nested lo ops ov er each of these v ariables i 1 , i 2 , . . . i n etc. where each of them go es from 0 to n − 1. This will allow us to go through each of the output click patterns | i 1 i 2 . . . i n ⟩ . 5. T o av oid going o v er conﬁgurations [ i 1 , i 2 , . . . , i n ] for whic h I = P n j =1 i j ≥ n , w e introduce the condition in the k -th lo op for 2 ≤ k ≤ n that chec ks whether k X j =1 i j = n (22) 11 If this condition is satisﬁed, then we hav e a break command, so that we break out of the i k lo op and jump to the next iteration of i k − 1 . T o express this more explicitly , we hav e i 1 going from 0 to n − 1. Then inside this, a second lo op runs o v er i 2 from 0 to n − 1. Inside that, we chec k whether i 1 + i 2 = n . If this is true, then w e execute the break command for the lo op o ver i 2 . If this condition is not satisﬁed, then w e go in to the next lo op o ver i 3 going from 0 to n − 1. Then we chec k whether i 1 + i 2 + i 3 = n . If true, then break, otherwise we go in to the i 4 lo op, and so on. The purp ose of this set of break conditions is that w e kno w that there is zero success rate for any click patterns with n total photons, and there are no click patterns with more than n photons. Therefore, we only need to calculate the probabilities for the cases where I < n . 6. In the innermost lo op, which in volv es i n running from 0 to n − 1, w e assign I = i 1 + i 2 + . . . + i n . Then, as describ ed in the previous step, we chec k whether I = n and apply a break condition if this is true as described ab o v e. But in the “else” case, calculate the probabilities P ± ,i 1 i 2 ...i n for the click pattern | i 1 i 2 . . . i n ⟩ based on equations ( 4 ) (by employing the T aylor series co eﬃcien t command in a mathematical softw are) and ( 5 ). W e p erform these calculations by taking ϕ = 0 for conv enience, as we hav e sho wn that the results are the same for all v alues of ϕ . 7. we then apply the condition that if P + ,i 1 i 2 ...i n P − ,i 1 i 2 ...i n  = 0 (i.e. if b oth are simultaneously non-zero), then f n, I + = f n, I + + P + ,i 1 i 2 ...i n , f n, I − = f n, I − + P − ,i 1 i 2 ...i n . (23) That is, failure for measuring b oth | + ⟩ and |−⟩ . 8. In the “else” case, i.e. P + ,i 1 i 2 ...i n P − ,i 1 i 2 ...i n = 0, w e execute the commands s n, I + = s n, I + + P + ,i 1 i 2 ...i n , s n, I − = s n, I − + P − ,i 1 i 2 ...i n . (24) This wa y , whichev er one of P + ,i 1 i 2 ...i n and P − ,i 1 i 2 ...i n is non-zero gets added to the corresponding success rate, while the other one whic h is zero will naturally add nothing. If b oth are zero, then again, these commands will add nothing to the success rate v ariables s n, I ± . 9. W e now hav e all the success and failure probabilities s n, I ± and f n, I ± for I total photon clic ks for all I ∈ { 0 , 1 , . . . , n } . W e just need to add them to obtain the total success and failure rates. s n ± = n X I =0 s n, I ± , f n ± = n X I =0 f n, I ± . (25) 10. T o make sure that our co de is working correctly , we can apply the consistency c hecks s n, I ± + f n, I ± = P n, I , s n ± + f n ± = 1 . (26) 12 Algorithm 1: Calculation of Success and F ailure Probabilities Initialize s n, I ± ← 0 and f n, I ± ← 0, I ∈ { 0 , 1 , . . . , n } ; f n, 0 ± ← P n, 0 ; f n,n ± ← P n,n ; Deﬁne index array [ i 1 , i 2 , . . . , i n ]; for i 1 ← 0 to n − 1 do for i 2 ← 0 to n − 1 do if i 1 + i 2 = n then break ; for i 3 ← 0 to n − 1 do if i 1 + i 2 + i 3 = n then break ; . . . for i n ← 0 to n − 1 do I ← i 1 + i 2 + · · · + i n ; if I = n then break ; Calculate P ± , i 1 i 2 ...i n using Eqs. ( 4 ) and ( 5 ); if P + , i 1 i 2 ...i n · P − , i 1 i 2 ...i n  = 0 then f n, I + ← f n, I + + P + , i 1 i 2 ...i n ; f n, I − ← f n, I − + P − , i 1 i 2 ...i n ; else s n, I + ← s n, I + + P + , i 1 i 2 ...i n ; s n, I − ← s n, I − + P − , i 1 i 2 ...i n ; end end end end end s n ± ← n X I =0 s n, I ± ; f n ± ← n X I =0 f n, I ± ; Consistency c hecks: ; s n, I ± + f n, I ± = P n, I ; s n ± + f n ± = 1; While the ab o v e-mentioned pro cedure works in principle, the calculations scale exp onentially with n and take increasingly longer time. The diﬃcult y arises from the fact that the co eﬃcients for the v arious terms in the p olynomial ( 4 ) ha ve many con tributions, so adding o v er them tak es a lot of operations. One p ossible wa y to simplify this pro cess is to set any creation op erators for which there are no photons in a given click pattern to b e zero. Then the p olynomial simpliﬁes a bit and the co eﬃcien ts require somewhat fewer operations, though the ov erall calculation still scales exp onen tially . In our w ork, w e hav e em plo y ed this approach and this has allow ed us to calculate a few more results than what would otherwise hav e b een p ossible. Results broken down in terms of total photon coun ts I Here are the patterns w e observ e based on the break-down of our results for the success and failure rates in terms of the total num ber of photons in the v arious output click patterns. 1. Click patterns with 0 or n photons in the output give zero success in measuring | + ϕ ⟩ or |− ϕ ⟩ . That is, s n, 0 ± = s n,n ± = 0. 2. W e get a 0 success rate for measuring | + ϕ ⟩ except when we hav e I = n/ 2 photons in the output click patterns. Th us, for o dd n , for which an integer n/ 2 do es not exist, the total success rate for | + ϕ ⟩ is zero. 3. F or n/ 2 photons in the output click patterns, we get p erfect discrimination b et ween | + ϕ ⟩ and |− ϕ ⟩ with the 13 success probabilities s n, n 2 ± = P n, n 2 = 1 2 n  n n 2  , (27) where in s n,n/ 2 ± , we are using the notation s n, I ± , in which I is the total num b er of photons in the output click patterns b eing considered which in this case is I = n/ 2. Note that this decreases as n gets larger and go es to 0 as n → ∞ . This explains why the success rate for | + ϕ ⟩ go es to zero in the large n limit. 4. The total success probability for measuring |− ϕ ⟩ from click patterns with a total of I photons is s n, I − = 4( n − I ) I  n I  2 n n 2 (28) Although we do not ha ve a formal pro of for this formula for all n and I , we describ e how we obtained it in the next section. W e should mention here that it is a simple exercise to see that this reduces to ( 27 ) when we take I = n/ 2. While w e do not hav e a rigorous pro of for any of the ab ov e expressions, we show in the next section how we can obtain them. Expressions for the total success rates for I photon clic k patterns W e will now sho w how w e obtain the expressions for the success rates s n, I ± that we observe and report in the previous section. F or con venience, we will w ork this out for the ϕ = 0 case, so that |± ϕ ⟩ = |±⟩ = [ | 0 ⟩ ± | 1 ⟩ ] / √ 2, but our results will b e applicable for all ϕ as we hav e prov ed in the main text. Let | + I ⟩ and |− I ⟩ b e the (normalized) pro jections of | + 1 + 2 . . . + n ⟩ and | − 1 + 2 . . . + n ⟩ , respectively , on to the subspace of states where I qubits are in | 1 ⟩ , and the res t are in | 0 ⟩ : | + I ⟩ = 1 q  n I  1 X i 1 ,i 2 ,...,i n =0 , i 1 + i 2 + ... + i n = I | i 1 i 2 i 3 . . . i n ⟩ |− I ⟩ = 1 q  n I  1 X i 1 ,i 2 ,...,i n =0 , i 1 + i 2 + ... + i n = I ( − 1) i 1 | i 1 i 2 i 3 . . . i n ⟩ (29) Let the unitary transformation asso ciated with our linear interferometer b e U lin , so that the | + I ⟩ and |− I ⟩ states going in to the interferometer give the output states | + ′ I ⟩ = U lin | + I ⟩ |− ′ I ⟩ = U lin |− I ⟩ (30) T o measure our single-rail qubit in the X basis, w e need to discriminate betw een the states | + ′ I ⟩ and |− ′ I ⟩ for each I . No w, consider the case when these hav e the forms | + ′ I ⟩ = I X i 1 ,i 2 ,...,i n =0 , i 1 + i 2 + ... + i n = I d + ,i 1 i 2 ...i n | i 1 i 2 . . . i n ⟩ |− ′ I ⟩ = I X i 1 ,i 2 ,...,i n =0 , i 1 + i 2 + ... + i n = I ( γ n, I d + ,i 1 i 2 ...i n + d − ,i 1 i 2 ...i n ) | i 1 i 2 . . . i n ⟩ (31) with 0 ≤ | γ n, I | ≤ 1, and d + ,i 1 i 2 ...i n d − ,i 1 i 2 ...i n = 0 for any giv en | i 1 i 2 . . . i n ⟩ conﬁguration i.e., one of d + ,i 1 i 2 ...i n and d − ,i 1 i 2 ...i n is zero. This wa y , when γ n, I  = 0, the output photon clic k patterns | i 1 i 2 . . . i n ⟩ in | + ′ I ⟩ also app ear in |− ′ I ⟩ , whereas |− ′ I ⟩ also has some click patterns not in | + ′ I ⟩ . But when γ n, I = 0, then | + ′ I ⟩ and |− ′ I ⟩ give totally non-o verlapping click patterns, represen ting p erfect distinguishability b etw een the single-rail qubit b eing found to b e in | + ⟩ and |−⟩ when the total num b er of photons in the obtained photon click pattern is I . 14 No w, if |± ′ I ⟩ ha ve the forms ( 31 ), then given that we ha v e a total of I photons in an output click pattern, the conditional success probability for correctly measuring our single-rail qubit to b e in | + ⟩ will b e 0 when γ n, I  = 0, and 1 when γ n, I = 0. And for measuring the single-rail qubit to b e in |−⟩ , we will hav e a conditional failure probability of γ 2 n, I , which will translate in to a success rate of 1 − γ 2 n, I . Multiplying these conditional probabilities by the probability P n, I that we hav e a click pattern with i total photons then gives the ov erall success rates for correctly measuring our single-rail qubit to b e in | + ⟩ and |−⟩ : s n + = 0 for γ n, I  = 0 s n + = P n, I for γ n, I = 0 s n − = P n, I  1 − γ 2 n, I  , (32) with γ n, I = ⟨ + ′ I |− ′ I ⟩ = ⟨ + I |− I ⟩ , where we hav e used the deﬁnitions ( 30 ). It is also straigh tforward to argue the conv erse of the ab ov e: if the success and failure rates hav e the form ( 34 ), then it means the states | + ′ I ⟩ and |− ′ I ⟩ ha v e the forms ( 31 ). It turns out that our success and failure rates for | + ⟩ and |−⟩ for the p ow er-of-2 Hadamard co des and QFTs that w e ha v e work ed out, indeed ha ve the ab ov e forms. This can for instance b e seen clearly in the 2- and 4-comp onent Hadamard co de cases explicitly work ed out in the next t wo sections. What remains to b e done is to calculate γ n, I . Recalling ( 29 ), note that | + I ⟩ has a total of  n I  terms, all of which ha ve plus signs. On the other hand, |− I ⟩ has a total of  n I  terms, out of which  n − 1 I − 1  terms hav e minus signs, and the remaining  n I  −  n − 1 I − 1  terms hav e p ositive signs. The inner product ⟨ + I |− I ⟩ is the num b er of p ositiv e terms min us the num ber of negative terms, up to the ov erall 1 /  n I  factor arising from the normalization factors in | + I ⟩ and |− I ⟩ . W e thus obtain γ n, I = ⟨ + I |− I ⟩ = 1  n I   n I  − 2  n − 1 I − 1  = n − 2 I n (33) Note that this is zero when I = n/ 2, giving us perfect discrimination b etw een | + ⟩ and |−⟩ . Inserting this in to ( 34 ) along with P n, I =  n I  / 2 n from ( 18 ), we then obtain the success rates s n, I + = 0 for I  = n 2 s n, I + =  n I  2 n for I = n 2 s n, I − = 4( n − I ) I  n I  2 n n 2 (34) This is the expression we rep orted in ( 28 ) in the previous section. The 2-comp onent Green Mac hine, i.e. a 50-50 b eam splitter Here w e describe the explicit analytical calculations of the n = 2 case for an X basis measuremen t where w e use | + ⟩ ancillas. T aking ϕ = 0, recall that the input states | ± 1 + 2 ⟩ are created by the op erators 1 2 (1 + a † 1 )(1 + a † 2 ) = 1 2 (1 + a † 1 + a † 2 + a † 1 a † 2 ) | 0 1 0 2 ⟩ 1 2 (1 + a † 1 )(1 + a † 2 ) = 1 2 (1 − a † 1 + a † 2 − a † 1 a † 2 ) (35) No w, consider feeding these into a 50-50 b eam splitter whic h replaces a † 1 and a † 2 b y the output creation op erators a † 1 7→ ( b † 1 + b † 2 ) / √ 2 a † 2 7→ ( b † 1 − b † 2 ) / √ 2 (36) 15 After a bit of simpliﬁcation, we obtain 1 2 (1 + a † 1 )(1 + a † 2 ) 7→ 1 2  1 + √ 2 b † 1 + [( b † 1 ) 2 − ( b † 2 ) 2 ]  1 2 (1 − a † 1 )(1 + a † 2 ) 7→ 1 2  1 − √ 2 b † 2 − [( b † 1 ) 2 − ( b † 2 ) 2 ]  (37) These tell us that w e obtain the clic k patterns | 1 1 0 2 ⟩ and | 0 1 1 2 ⟩ from the input states | + 1 + 2 ⟩ and | − 1 + 2 ⟩ , respectively , with probabilit y 1 / 2. These clic k patterns give 50% success rate. The zero or tw o-photon clic k patterns corresp ond to failure of measuring our single-rail qubit in the X basis. The 4-comp onent Green Mac hine W e no w show the explicit analytical calculations of the n = 4 case for an X basis measuremen t using | + ⟩ ancillas and a 4-comp onent Green Mac hine. W e again start with the op erators that act on the v acuum to generate the | ± 1 + 2 + 3 + 4 ⟩ input states: 1 4 (1 ± a † 1 )(1 + a † 2 )(1 + a † 3 )(1 + a † 4 ) (38) W e then apply the ﬁrst b eam splitter la yer of the 4-component Green Mac hine according to the re cursiv e construction describ ed earlier. In this lay er, one b eam splitter mixes p orts 1 and 2 with the transformations a † 1 7→ 1 √ 2 ( c 1 + c 2 ) , a † 2 7→ 1 √ 2 ( c 1 − c 2 ) , (39) and the second b eam splitter mixes p orts 3 and 4 with the transformations a † 3 7→ 1 √ 2 ( c 3 + c 4 ) , a † 4 7→ 1 √ 2 ( c 3 − c 4 ) . (40) Here, the c i op erators are creation op erators for p orts num ber i = 1 , . . . , 4, and w e hav e tak en the lib erty not to put daggers on them in order to simplify the notation. W e can aﬀord to tak e this liberty since we do not ha v e an y annihilation op erators an ywhere in this en tire analysis, and will therefore take the daggers to be implicit in the rest of this section. No w, for the input state | + 1 + 2 + 3 + 4 ⟩ , we get 1 4 [1 + √ 2 b 1 + ( b 2 1 − b 2 2 ) / 2][1 + √ 2 b 3 + ( b 2 3 − b 2 4 ) / 2] (41) F or the input state | − 1 + 2 + 3 + 4 ⟩ , we get 1 4 [1 − √ 2 b 2 − ( b 2 1 − b 2 2 ) / 2][1 + √ 2 b 3 + ( b 2 3 − b 2 4 ) / 2] (42) No w, apply the second la y er of beam splitters. One b eam splitter mixes p orts 1 and 3, and another one mixes ports 2 and 4. The transformations are c 1 7→ 1 √ 2 ( d 1 + d 3 ) , c 3 7→ 1 √ 2 ( d 1 − d 3 ) (43) for one b eam splitter, and c 2 7→ 1 √ 2 ( d 2 + d 4 ) , c 4 7→ 1 √ 2 ( d 2 − d 4 ) (44) 16 for the other b eam splitter, with d j the photon creation op erators for the output of the 4-comp onen t Green Machine for j ∈ { 1 , . . . , 4 } . F or the input state | + 1 + 2 + 3 + 4 ⟩ going into our 4-comp onen t Green Machine, we get 1 4  1 + ( d 1 + d 3 ) + 1 4 (( d 1 + d 3 ) 2 − ( d 2 + d 4 ) 2 )   1 + ( d 1 − d 3 ) + 1 4 (( d 1 − d 3 ) 2 − ( d 2 − d 4 ) 2 )  = 1 4  1 + 2 d 1 + 1 2 (3 d 2 1 − d 2 2 − d 2 3 − d 2 4 ) + 1 4 ( d 1 − d 3 )[( d 1 + d 3 ) 2 − ( d 2 + d 4 ) 2 ] + 1 4 ( d 1 + d 3 )[( d 1 − d 3 ) 2 − ( d 2 − d 4 ) 2 ] + 1 16 [( d 1 + d 3 ) 2 − ( d 2 + d 4 ) 2 ][( d 1 − d 3 ) 2 − ( d 2 − d 4 ) 2 ]  . (45) F or | − 1 + 2 + 3 + 4 ⟩ , we get 1 4  1 − ( d 2 + d 4 ) − 1 4 (( d 1 + d 3 ) 2 − ( d 2 + d 4 ) 2 )   1 + ( d 1 − d 3 ) + 1 4 (( d 1 − d 3 ) 2 − ( d 2 − d 4 ) 2 )  = 1 4  1 + d 1 − d 2 − d 3 − d 4 − d 1 ( d 2 + d 3 + d 4 ) + d 2 d 3 + d 2 d 4 + d 3 d 4 − 1 4 ( d 1 − d 3 )[(( d 1 + d 3 ) 2 − ( d 2 + d 4 ) 2 ] − 1 4 ( d 2 + d 4 )[( d 1 − d 3 ) 2 − ( d 2 − d 4 ) 2 ] − 1 16 [( d 1 + d 3 ) 2 − ( d 2 + d 4 ) 2 ][( d 1 − d 3 ) 2 − ( d 2 − d 4 ) 2 ]  . (46) No w, w e compare ( 45 ) and ( 46 ) line by line. In the linear terms, we see that for | + ⟩ we hav e d 1 / 2, and d 1 / 4 also app ears in the |−⟩ case. Th us for 1-photon states, w e ha v e total failure for | + ⟩ , and a failure rate of (1 / 4) 2 = 1 / 16 for |−⟩ . Since the probabilit y for obtaining one of the click patterns with 1- total photon click is P 4 , 1 =  4 1  / 2 4 = 1 / 4, w e obtain a success rate of s 4 , 1 − = 1 4 − 1 16 = 3 16 (47) for correctly measuring |−⟩ from 1-photon clic k patterns, whic h matches the formula ( 28 ) w e ga ve while describing the general pattern in terms of the photon counts in the v arious click patterns. F or click patterns with 2 photons, w e compare the quadratic terms. Note that there are no common quadratic terms in ( 45 ) and ( 46 ). This means that we hav e full success from 2-photon click patterns: s 4 , 2 ± = P 4 , 2 =  4 2  2 4 = 3 8 (48) Again, this matches the results ( 27 ) while describing the general patterns in terms of the break down in terms of the total photon counts in the output click patterns. Next, we come to the qubit terms. With a bit of simpliﬁcation, the qubit terms in ( 45 ) simplify to 1 8 d 3 1 − 1 8 d 1 ( d 2 2 + d 2 3 + d 2 4 ) + 1 4 d 2 d 3 d 4 (49) In the qubit terms in ( 46 ), it is clear that the ( d 2 + d 4 )[( d 1 − d 3 ) 2 − ( d 2 − d 4 ) 2 ] / 16 part will ha ve nothing o verlapping with this. Ho wev er, if we simplify − ( d 1 − d 3 )[( d 1 + d 3 ) 2 − ( d 2 + d 4 ) 2 ] / 16, then we ﬁnd that it has the part − 1 16 d 3 1 + 1 16 d 1 ( d 2 2 + d 2 3 + d 2 4 ) − 1 8 d 2 d 3 d 4 (50) plus some additional uncommon terms. Thus, everything in the qubit part for the | + ⟩ state for our single-rail qubit is also app earing in the output state corresp onding to |−⟩ , and hence we hav e zero probability of success for correctly measuring | + ⟩ from 3-photon clic k patterns. F or |−⟩ , we ha ve half of the p olynomial that appears in | + ⟩ , so the failure probability is P 4 , 3 / 4, with the factor of 1 / 4 arising from squaring 1 / 2. Thus the total success rate associated with 3-photon click patterns is s 4 , 3 − = P 4 , 3 − P 4 , 3 4 = 3 16 (51) Again, this is consistent with the formula ( 28 ) from the results we obtained in terms of the photon counts. 17 Com bining these results, the ov erall success probabilities are s 4+ = 0 + 3 8 + 0 = 3 8 s 4 − = 3 16 + 3 8 + 3 16 = 3 4 s 4 , ov erall = 1 2 ( s 4+ + s 4 − ) = 9 16 (52) Non-p o w er of 2 Hadamard unitaries and results for the 12-component case While p ow er of 2 Hadamard unitaries ha ve the ab ov e-men tioned b eautiful and recursive form, Hadamard matrices ha ve also b een sho wn to exist for several n umbers other than p o wers of 2 with the general conjecture that these exist for all multiples of 4 [ 33 ]. F or example, one construction for n = 12 gives the unitary transformation matrix U GM , 12 ≡ 1 √ 12                     1 1 1 1 1 1 1 1 1 1 1 1 1 − 1 1 − 1 1 1 1 − 1 − 1 − 1 1 − 1 1 − 1 − 1 1 − 1 1 1 1 − 1 − 1 − 1 1 1 1 − 1 − 1 1 − 1 1 1 1 − 1 − 1 − 1 1 − 1 1 − 1 − 1 1 − 1 1 1 1 − 1 − 1 1 − 1 − 1 1 − 1 − 1 1 − 1 1 1 1 − 1 1 − 1 − 1 − 1 1 − 1 − 1 1 − 1 1 1 1 1 1 − 1 − 1 − 1 1 − 1 − 1 1 − 1 1 1 1 1 1 − 1 − 1 − 1 1 − 1 − 1 1 − 1 1 1 1 1 1 − 1 − 1 − 1 1 − 1 − 1 1 − 1 1 − 1 1 1 1 − 1 − 1 − 1 1 − 1 − 1 1 1 1 − 1 1 1 1 − 1 − 1 − 1 1 − 1 − 1                     . (53) Note that this is not a symmetric matrix and its transp ose will also give an equally v alid normalized Hadamard matrix. Thus, non p ow er-of-2 Hadamard matrices do not hav e the nice symmetries that the p o wer-of-2 cases hav e. Nev ertheless, as an example of the non-p ow er of 2 case, w e hav e calculated the success and failure rates for the ab o v e-mentioned 12-comp onent unitary transformation. W e ﬁnd that this case substantially deviates from the QFT and the p o w er of 2 Hadamard co des and gives a m uc h lo w er o verall success rate of ab out 36 . 63%. This deviation is not entirely surprising considering that, unlike p o w er-of-2 Hadamard co des or QFTs, the 12-comp onent Hadamard matrix is not so nice and symmetric. F or details, the ov erall success and failure rates for measuring | + ⟩ and |−⟩ are s 12+ = 6731395 47775744 ≈ 0 . 140956604 s 12 − = 6106045627 10319560704 ≈ 0 . 5916962749 f 12+ = 41044349 47775744 ≈ 0 . 8591043396 f 12 − = 4213515077 10319560704 ≈ 0 . 4083037251 (54) The break-up in terms of the total photons i in the output click-patterns is given in table I . Using coherent state ancillas instead of | + ⟩ As before, let us label the optical mo de enco ding our single-rail qubit, whic h w e wish to measure in the X basis, as mo de 1, and consider mixing it with a coherent state ancilla | α ⟩ in mo de 2, using a ﬁft y-ﬁfty b eam splitter of the form ( 16 ) [ 47 ]. W e will denote the coherent state in mo de 2 as | α 2 ⟩ with the subscript 2 indicating the mo de num ber. 18 T ABLE I. V alues of s 12 , I + and s 12 , I − No. of photons I s 12 , I + s 12 , I − 0 0 0 1 0 11 / 12288 2 0 55 / 6144 3 0 1375 / 36864 4 0 605 / 6912 5 0 7535 / 55296 6 121385 / 884736 16093 / 110592 7 0 210595 / 1990656 8 0 3685 / 73728 9 774455 / 214990848 2291245 / 143327232 10 9295 / 143327232 1817585 / 573308928 11 6325 / 214990848 3182113 / 10319560704 12 0 0 Note that if we mix the coherent state | α 2 ⟩ with the F o ck states | 0 ⟩ or | 1 ⟩ in the ﬁrst mo de, we obtain | 0 1 α 2 ⟩ →     α/ √ 2  1 ,  − α/ √ 2  2 E | 1 1 α 2 ⟩ → 1 √ 2 ( b † 1 + b † 2 )     α/ √ 2  1 ,  − α/ √ 2  2 E (55) Here, 0 1 and 1 1 on the LHS denote the 0 and 1-photon F ock states in input mode 1, and b † 1 and b † 2 are the creation op erators for the t wo output modes of the b eam splitter, which we still lab el as mo des 1 and 2. T o successfully measure the input state of mo de 1 in the |±⟩ basis, we need b oth the ab o ve terms to hav e the same total num ber of photons in order to scram ble the information ab out whether w e had an incoming photon in mo de 1 or not. Putting the ab ov e t wo equations together as ( | 0 1 α 2 ⟩ ± | 1 1 α 2 ⟩ ) / √ 2, and expanding the coheren t states in terms of the creation op erators and F o ck states, this condition gives the terms e −| α | 2 / 2 ( − 1) j √ 2  1 i ! j ! ( α/ √ 2) i + j ± ( − 1) j √ 2  1 ( i − 1)! j ! − 1 i !( j − 1)!  ( α/ √ 2) i + j − 1  ( b † 1 ) i ( b † 2 ) j | 0 1 0 2 ⟩ (56) = e −| α | 2 / 2 ( − 1) j √ 2 i ! j ! ( α/ √ 2) i + j [1 ± ( i − j ) /α ] | i 1 j 2 ⟩ (57) This gives the condition α = ± ( i − j ) for a successful X basis measurement of mode 1. Assuming p ositive v alues of α , the click patterns | i, i − α ⟩ and | i − α , i ⟩ mean the signal-rail qubit b eing successfully measured and found to b e in | + ⟩ and |−⟩ , resp ectiv ely , both with the probability P s, 2 ,i,α = 2 e − α 2 i !( i − α )! ( | α | 2 / 2) 2 i − α , | α | = 1 , 2 , . . . , and i = | α | , | α | + 1 , | α | + 2 , . . . (58) where the subscript s in P s, 2 ,i,α denotes the fact that this is the probabilit y of success, and 2 indicates that w e are applying a balanced b eam splitter, i.e., a 2-comp onen t Green Machine. The total probability of success is thus P s, 2 ,α = ∞ X i = α P s, 2 ,i,α = 2 e − α 2 I | α |  α 2  , α = ± 1 , ± 2 , . . . (59) where I | α |  α 2  is the mo diﬁed Bessel function of the ﬁrst kind of order | α | with argument α 2 . This gives a maximum success rate of approximately 0 . 4158 for | α | = 1 and decreasing success rates with increasing | α | . Note that replacing α with − α gives the same success rates, with the only c hange being that the click patterns asso ciated with the | + ⟩ and |−⟩ outcomes are switched. 19 This metho d of course requires a ﬁne-tuned v alue of α , and is therefore prone to error. In the context of loading a single-rail qubit in some arbitrary state | ψ 1 ⟩ ≡ υ | 0 1 ⟩ + ξ | 1 1 ⟩ on to a quantum memory , we can ask what actual state is loaded onto the latter if α is not p erfectly ﬁne-tuned to the correct v alue. Recalling the state transfer proto col from the motiv ation section of the supplementary material, we initialize the quantum memory in | 0 a ⟩ and apply a CNOT gate from our single-rail qubit to this memory qubit. But now use a coherent state | α ⟩ in mo de 2. The joint state of the tw o optical modes and the quantum memory after the CNOT operation but b efore the b eam splitter is ( υ + ξ a † 1 X a ) | 0 1 α 2 0 a ⟩ , (60) where X a is a Pauli X op erator on the quantum memory . Recall that we need the coherent state to pro vide one additional photon when mixing with | 0 1 ⟩ than it do es when mixing with | 1 1 ⟩ for a successful X basis measuremen t. Applying this condition and rep eating the steps inv olved in deriving equation ( 57 ), we ﬁnd that the state after mixing the tw o optical modes in a balanced b eam splitter is ( − 1) j e −| α | 2 / 2 1 √ i ! j ! ( α/ √ 2) i + j [ αυ + ( i − j ) ξ X a ] /α | i 1 j 2 0 a ⟩ (61) = ( − 1) j e −| α | 2 / 2 α 1 √ i ! j ! ( α/ √ 2) i + j p | αυ | 2 + ( i − j ) 2 | ξ | 2 αυ + ( i − j ) ξ X a p | αυ | 2 + ( i − j ) 2 | ξ | 2 | i 1 j 2 0 a ⟩ , (62) where we are using the same n um b ers to label the input and output mo des of the b eam splitter. Th us, if we obtain the click pattern | i 1 j 2 ⟩ in a photon n um b er detection of the b eam splitter outputs, the quan tum memory ends up in the state | ψ loaded ⟩ ≡ 1 p | αυ | 2 + ( i − j ) 2 | ξ | 2 [ αυ | 0 a ⟩ + ( i − j ) ξ | 1 a ⟩ ] (63) with the probability P loading ,α, i , j = e −| α | 2 | α | 2 1 i ! j ! ( | α | 2 / 2) i + j  | αυ | 2 + ( i − j ) 2 | ξ | 2  . (64) Note that the click pattern with the v alues of i and j in terchanged, has the same probability and gives the same state as | ψ loaded ⟩ , but with a ‘ − ’ sign on the | 1 a ⟩ term. Since this sign can b e ﬂipp ed with a Z gate, w e thus again obtain | ψ loaded ⟩ with the same probability . The combined probability of obtaining | ψ loaded ⟩ on obtaining the click patterns | i 1 j 2 ⟩ and | j 1 i 2 ⟩ is then 2 P loading ,α, i , j . And the total probability of this is the sum ov er all v alues of i and j for a ﬁxed | i − j | . Since the highest success rate for a perfect X basis measuremen t of the single rail qubit was obtained for | i − j | = 1 with α = ± 1, we in terpret the clic k patterns | i 1 ( i − 1) 2 ⟩ and | ( i − 1) 1 i 2 ⟩ as representing successful state loading, alb eit with p ossible imp erfection due to the v alue of α not b eing p erfectly ﬁne-tuned. The total success probabilit y of loading the single-rail qubit on to a quantum memory is then P tot ,α ≡ ∞ X i =1 2 P loading ,α, i , i − 1 = 2 e −| α | 2 | α | 2 ∞ X i =1 1 i !( i − 1)! ( | α | 2 / 2) 2 i − 1 [ | αυ | 2 + | ξ | 2 ] . (65) Next, let us consider employing 3 coherent state ancillas | α ⟩ and mixing them with our single-rail qubit in a 4- comp onen t Green Mac hine to p erform an X basis measurement. W e feed the single-rail qubit into the ﬁrst input, and the ancillas into the rest. Recall from the recursive construction of a Green Machine describ ed in section , that a 4-comp onen t Green Machine ﬁrst inv olves tw o separate balanced b eam splitters of the form ( 16 ) mixing mo de 1 with 2, and 3 with 4, and in step t w o, we mix mode 1 with 3, and 2 with 4 using similar 50-50 b eam splitters. Consequently , w e obtain | 0 1 α 2 α 3 α 4 ⟩ → | (3 α/ 2) 1 , ( − α/ 2) 2 , ( − α/ 2) 3 , ( − α/ 2) 4 ⟩ , | 1 1 α 2 α 3 α 4 ⟩ → 1 2 ( b † 1 + b † 2 + b † 3 + b † 4 ) | (3 α/ 2) 1 , ( − α/ 2) 2 , ( − α/ 2) 3 , ( − α/ 2) 4 ⟩ . (66) Here, as in the case of a single coheren t state ancilla, 0 1 and 1 1 denote the 0 and 1-photon F o c k states in the ﬁrst input mo de of the Green Machine, and b † 1 , b † 2 , b † 3 and b † 4 are the creation op erators for the output mo des of the 4-comp onen t 20 Green Machine. In the second equation, we hav e used the fact that the creation operator a † 1 for a photon going into input mo de 1 goes to b † 1 + b † 2 + b † 3 + b † 4 2 under the action of the Green Machine. No w, again, the coherent states need to provide 1 additional photon to mix with the | 0 1 ⟩ state compared to | 1 1 ⟩ to scram ble the information ab out whether we hav e | 0 1 ⟩ or | 1 1 ⟩ and provide a successful X basis measuremen t. Putting together the ab ov e tw o equations as | ± 1 α 2 α 3 α 4 ⟩ and applying this condition now giv es the terms in the output state exp( − 3 | α | 2 / 2) 1 √ 2 i ! j ! k ! l ! 3 i ( α/ 2) i + j + k + l [1 ± ( i/ 3 − j − k − l ) /α ] | i 1 j 2 k 3 l 4 ⟩ , (67) where | i 1 j 2 k 3 l 4 ⟩ means the output photon click pattern with i , j , k and l photons in mo des 1, 2, 3 and 4, resp ectiv ely . W e thus hav e the conditions α = i/ 3 − j − k − l and α = − ( i/ 3 − j − k − l ) for a successful measurement in | + 1 ⟩ and |− 1 ⟩ , resp ectiv ely . The success rates are thus P s ± , 4 ,α = 2 exp( − 3 | α | 2 ) ∞ X i,j,k,l =0 ,i/ 3 − j − k − l = ± α 9 i i ! j ! k ! l ! ( | α | 2 / 4) i + j + k + l . (68) Here, the ± in the subscript of P s ± , 4 ,α indicates whether it is the success rate for measuring | + ⟩ or |−⟩ , and 4 and α indicate that this is for a 4-comp onen t Green Mac hine with coherent states | α ⟩ used as the three ancillas. W e can see that the success rates for | + ⟩ and |−⟩ are asymmetric. It turns out that the maximum success rate is obtained for α = 1 / 3, and is ab out P s +4 ,α = 0 . 358 for measuring | + ⟩ , but this gives a 0 . 0037 success rate for measuring |−⟩ , so the a verage success rate, assuming equal priors for both |±⟩ , is therefore only ab out 18 . 10%.. F or α = ± 1, we obtain a success rate of only 0 . 0544 for measuring |−⟩ , and 0 for measuring | + ⟩ . In short, the p erformance of the 4-comp onen t Green Machine with three coheren t state ancillas is muc h low er than a balanced b eam splitter with just one coheren t state ancilla. Lo oking at the falling trend for the success rate as we go from the n = 2 case to the n = 4 one, we do not explicitly consider larger Green Machines with coheren t state ancillas b eyond the n = 4 case, but it is a straightforw ard exercise to generalize the success rate formula ( 68 ) and obtain P s ± ,n,α = 2 exp( − ( n − 1) | α | 2 ) ∞ X i 1 ,i 2 ,,...i n =0 ,i 1 / ( n − 1) − i 2 − i 3 − ... − i n = ± α ( n − 1) 2 i 1 i 1 ! i 2 ! . . . i n ! ( | α | 2 /n ) i 1 + i 2 + ... + i n . (69) Here, we are using i 1 , i 2 , . . . i n to lab el the num b ers of photons in mo des 1 , 2 , . . . , n , resp ectiv ely , so the subscripts are part of the diﬀerent sym b ols denoting the photon counts in the diﬀerent mo des, and not the mo de lab els. W e are switc hing to this notation instead of using separate letters as we did for the 2 and 4 comp onent cases to account for the general num b er of mo des.

Boosted linear-optical measurements on single-rail qubits with unentangled ancillas

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