Exponential quantum advantage for measuring fermionic operators

Abstract

Method, systems, and apparatus for measuring fermionic operators. In one aspect, a method includes obtaining an input comprising k-body Majorana operators, a predefined precision, and copies of a tensor product of a quantum state and the quantum state. For each Majorana operator, a basis to measure the Majorana operator in is determined. The input is processed to obtain expectation values that correspond to the Majorana operators. A graph is constructed by, for each non-zero expectation value, adding a vertex to the graph that represents a Majorana operator that corresponds to the non-zero expectation value and adding edges between vertices in the graph that represent anticommuting Majorana operators. A vertex coloring algorithm is applied to the graph. For a color that corresponds to the Majorana operator, a simultaneous eigenbasis of Majorana operators in the graph with the color is determined. The Majorana operator is measured in the determined basis.

Description

[0001]Attorney Docket: 56113-0625WO1 EXPONENTIAL QUANTUM ADVANTAGE FOR MEASURING FERMIONIC OPERATORS BACKGROUND This specification relates to quantum computing. Fermionic operators describe the quantum states and interactions of fermionic particles, such as electrons. Example fermionic operators include creation operators, which add a fermion to a specific quantum state, annihilation operators, which remove a fermion from a state., and the number operator, which quantifies the occupancy of a given quantum state. In quantum chemistry, fermionic operators are used to compute the electronic structure and molecular properties of molecular systems. SUMMARY This specification describes technologies for measuring elements of fermionic operators with a reduced quantum memory requirement. In general, one innovative aspect of the subject matter described in this specification can be implemented in a method performed by a quantum computer, the method comprising: obtaining an input comprising a list of M quadratic Majorana operators, a predefined precision, and N copies of a tensor product of a quantum state on n fermion modes with the quantum state on the n fermion modes; processing the input to obtain a list of expectation values, wherein each expectation value in the list corresponds to a respective quadratic Majorana operator; constructing a graph, comprising assigning a single Majorana modes included in the M quadratic Majorana operators to respective vertices in the graph and adding edges between vertices that represent a pair of single Majorana modes that correspond to a non-zero expectation value in the list, wherein pairs of single Majorana modes represent respective quadratic Majorana operators; applying an edge coloring algorithm to color the graph in multiple colors, wherein edges incident on a vertex have different colors; and measuring each quadratic Majorana operator in the list of M quadratic Majorana operators, comprising, for each quadratic Majorana operator, measuring multiple copies of the quantum state on n modes in a simultaneous eigenbasis of all quadratic Majorana operators in the graph with a same color as the quadratic Majorana operator to obtain an empirical estimate of the quadratic Majorana operator. In general, another innovative aspect of the subject matter described in this specification can be implemented in a method performed by a quantum computer, the method Attorney Docket: 56113-0625WO1 comprising: obtaining an input comprising a list of M quadratic Majorana operators, a predefined precision, and N copies of a tensor product of a quantum state on n fermion modes and the quantum state on the n fermion modes; for each quadratic Majorana operator in the list: determining a basis to measure the quadratic Majorana operator in, comprising: processing the input to obtain a list of expectation values, wherein each expectation value in the list corresponds to a respective quadratic Majorana operator; constructing a graph, comprising assigning single Majorana modes included in the M quadratic Majorana operators to respective vertices in the graph and adding edges between vertices that represent a pair of single Majorana modes that correspond to a non-zero expectation value in the list; applying an edge coloring algorithm to color the graph in multiple colors, wherein incident edges on a vertex have different colors; for a color that corresponds to the quadratic Majorana operator, determining a simultaneous eigenbasis of quadratic Majorana operators in the graph with the color; and measuring the quadratic Majorana operator in the determined basis to obtain an empirical estimate of the quadratic Majorana operator. Other implementations of these aspects include corresponding computer systems, apparatus, and computer programs recorded on one or more computer storage devices, each configured to perform the actions of the methods. A system of one or more quantum computers can be configured to perform particular operations or actions by virtue of having software, firmware, hardware, or a combination thereof installed on the system that in operation causes or cause the system to perform the actions. One or more computer programs can be configured to perform particular operations or actions by virtue of including instructions that, when executed by data processing apparatus, cause the apparatus to perform the actions. The foregoing and other implementations can each optionally include one or more of the following features, alone or in combination. In some implementations a magnitude of a difference between each empirical estimate of a quadratic Majorana operator and a trace of the quadratic Majorana operator with respect to the quantum state is less than or equal to the predefined precision. In some implementations an empirical estimate of a quadratic Majorana operator estimates a trace of the quadratic Majorana operator with respect to the quantum state. In some implementations measuring multiple copies of the quantum state on ^^ modes in a simultaneous eigenbasis of quadratic Majorana operators with the color to obtain an Attorney Docket: 56113-0625WO1 empirical estimate of each quadratic Majorana operator in the list of ^^ quadratic Majoranaoperators comprises measuring ^^ଶ ൌ ^^ ^୪୭^ெ ఢమ^copies of the quantum state. In some implementations colors comprise at most deg(Gε) + 1 colors, wherein deg(Gε) represents a of any vertex in the graph. In some implementations processing the input comprises: measuring each copy of the tensor product in an ^^-fold Bell basis to obtain a set of measurement outcomes, wherein each measurement outcome in the set corresponds to a respective tensor product and comprises a first binary string and a second binary string of length ^^; for each quadratic Pauli operator representation of a respective quadratic Majorana operator: computing a sum of values of a variable, each value in the sum corresponding to a respective measurement outcome, wherein the variable is equal to 1 if a first binary string included in a respective measurement outcome multiplied by a binary representation of a first Pauli operator included in the quadratic Pauli operator minus a second binary string included in a respective measurement outcome multiplied by a binary representation of a second Pauli operator included in the quadratic Pauli operator is equal to 0 modulo 2 or is equal to -1 if the first binary string included in the respective measurement outcome multiplied by the binary representation of the first Pauli operator included in the quadratic Pauli operator minus the second binary string included in the respective measurement outcome multiplied by the binary representation of the second Pauli operator included in the quadratic Pauli operator is equal to 1 modulo 2; determining whether the sum of values is less than or equal to a first predetermined threshold that depends on the predefined precision; and in response to determining that the sum of values is less than or equal to the first predetermined threshold that depends on the predefined precision, outputting the value 0; or in response to determining that the sum of values is greater than the first predetermined threshold that depends on the predefined precision, outputting a square root of the sum of values. In some implementations the first predetermined threshold is equal to 2 / 3 of a square of the predefined precision. In some implementations outputting the value 0 indicates that a magnitude of a trace of the corresponding quadratic Majorana operator with respect to the quantum state on ^^ fermion modes is less than or equal to the predefined precision. In some implementations outputting the square root of the sum of values indicates that a magnitude of a difference between a positive or negative square root of the sum of values and a trace of the corresponding quadratic Majorana operator with respect to the quantum Attorney Docket: 56113-0625WO1 state on n fermion modes is less than or equal to a second predetermined threshold that is based on the predefined precision. In some implementations the second predetermined threshold is equal to√ଶ ଶ√ଷ ^^ where ^^ represents the predefined precision. In some implementations processing the input to obtain a list of expectation valuesrequires ^^^ ൌ ^^ ^^^^ெ ఌర^ copies of the tensor product, wherein ^^ represents the predefined precision. In the method further comprises converting one or more of the empirical estimates of the quadratic Majorana operators to respective estimates of one or more creation or annihilation operators. In general, another innovative aspect of the subject matter described in this specification can be implemented in a method performed by a quantum computer, the method comprising: obtaining an input comprising a list of ^^ k-body Majorana operators, a predefined precision, and ^^ copies of a tensor product of a quantum state on ^^ fermion modes with the quantum state on the ^^ fermion modes; processing the input to obtain a list of expectation values, wherein each expectation value in the list corresponds to a respective k- body Majorana operator; constructing a graph, comprising, for each non-zero expectation value in the list, adding a vertex to the graph that represents a k-body Majorana operator that corresponds to the non-zero expectation value and adding edges between vertices in the graph that represent anticommuting k-body Majorana operators; applying a vertex coloring algorithm to color the graph in multiple colors, wherein adjacent vertices have different colors and a total number of colors used is minimized; measuring each k-body Majorana operator in the list of ^^ k-body Majorana operators, comprising, for each k-body Majorana operator, measuring multiple copies of the quantum state on ^^ modes in a simultaneous eigenbasis of all k-body Majorana operators in the graph with a same color as the k-body Majorana operator to obtain an empirical estimate of the k-body Majorana operator. In general, another innovative aspect of the subject matter described in this specification can be implemented in a method performed by a quantum computer, the method comprising: obtaining an input comprising a list of ^^ k-body Majorana operators, a predefined precision, and ^^ copies of a tensor product of a quantum state on ^^ fermion modes and the quantum state on the ^^ fermion modes; for each k-body Majorana operator in the list: determining a basis to measure the k-body Majorana operator in, comprising: processing the input to obtain a list of expectation values, wherein each expectation value in Attorney Docket: 56113-0625WO1 the list corresponds to a respective k-body Majorana operator; constructing a graph, comprising, for each non-zero expectation value in the list, adding a vertex to the graph that represents a k-body Majorana operator that corresponds to the non-zero expectation value and adding edges between vertices in the graph that represent anticommuting k-body Majorana operators; applying a vertex coloring algorithm to color the graph in multiple colors, wherein adjacent vertices have different colors and a total number of colors used is minimized; for a color that corresponds to the k-body Majorana operator, determining a simultaneous eigenbasis of k-body Majorana operators in the graph with the color; and measuring the k- body Majorana operator in the determined basis to obtain an empirical estimate of the k-body Majorana operator. Other implementations of these aspects include corresponding computer systems, apparatus, and computer programs recorded on one or more computer storage devices, each configured to perform the actions of the methods. A system of one or more quantum computers can be configured to perform particular operations or actions by virtue of having software, firmware, hardware, or a combination thereof installed on the system that in operation causes or cause the system to perform the actions. One or more computer programs can be configured to perform particular operations or actions by virtue of including instructions that, when executed by data processing apparatus, cause the apparatus to perform the actions. The foregoing and other implementations can each optionally include one or more of the following features, alone or in combination. In some implementations a magnitude of a difference between each empirical estimate of a k-body Majorana operator and a trace of the k-body Majorana operator with respect to the quantum state is less than or equal to the predefined precision. In some implementations an empirical estimate of a k-body Majorana operator estimates a trace of the k-body Majorana operator with respect to the quantum state. In some implementations measuring multiple copies of the quantum state on ^^ modes in a simultaneous eigenbasis of k-body Majorana operators with the color to obtain an empirical estimate of each k-body Majorana operator in the list of ^^ k-body Majoranaoperators comprises measuring ^^ ൌ ^^୪୭^ெ ଶ^^ copies of the quantum state. In some implementations of colors is less than or equal to a number of an order of ^ ఢమwhere ^^ represents the predefined precision. Attorney Docket: 56113-0625WO1 In some implementations processing the input comprises: measuring each copy of the tensor product in an ^^-fold Bell basis to obtain a set of measurement outcomes, wherein each measurement outcome in the set corresponds to a respective tensor product and comprises a first binary string and a second binary string of length ^^; for each quadratic Pauli operator representation of a respective k-body Majorana operator: computing a sum of values of a variable, each value in the sum corresponding to a respective measurement outcome, wherein the variable is equal to 1 if a first binary string included in a respective measurement outcome multiplied by a binary representation of a first Pauli operator included in the quadratic Pauli operator minus a second binary string included in a respective measurement outcome multiplied by a binary representation of a second Pauli operator included in the quadratic Pauli operator is equal to 0 modulo 2 or is equal to -1 if the first binary string included in the respective measurement outcome multiplied by the binary representation of the first Pauli operator included in the quadratic Pauli operator minus the second binary string included in the respective measurement outcome multiplied by the binary representation of the second Pauli operator included in the quadratic Pauli operator is equal to 1 modulo 2; determining whether the sum of values is less than or equal to a first predetermined threshold that depends on the predefined precision; and in response to determining that the sum of values is less than or equal to the first predetermined threshold that depends on the predefined precision, outputting the value 0; or in response to determining that the sum of values is greater than the first predetermined threshold that depends on the predefined precision, outputting a square root of the sum of values. In some implementations the first predetermined threshold is equal to 2 / 3 of a square of the predefined precision. In some implementations outputting the value 0 indicates that a magnitude of a trace of the corresponding k-body Majorana operator with respect to the quantum state on ^^ fermion modes is less than or equal to the predefined precision. In some implementations outputting the square root of the sum of values indicates that a magnitude of a difference between a positive or negative square root of the sum of values and a trace of the corresponding k-body Majorana operator with respect to the quantum state on n fermion modes is less than or equal to a second predetermined threshold that is based on the predefined precision. In some implementations the second predetermined threshold is equal to√ଶ ଶ√ଷ ^^ where ^^ represents the predefined precision. Attorney Docket: 56113-0625WO1 In some implementations processing the input to obtain a list of expectation valuesrequires ^^^ ൌ ^^ ^^^^ெ ఌర^ copies of the tensor product, wherein ^^ represents the predefined precision. In the method further comprises converting one or more of the empirical estimates of the k-body Majorana operators to respective estimates of one or more creation or annihilation operators. The subject matter described in this specification can be implemented in particular ways so as to realize one or more of the following advantages. The objective of fermionic learning is to learn properties of an unknown quantum state, e.g., a k-RDM of a fermionic system on n modes, using only a few physical copies of the quantum state. The non-commutative nature of local fermionic operators provides an obstruction to learning the k-RDM with single-copy measurements. This opens the door to the possibility of gaining a large advantage by using measurements which are entangled across multiple copies of the quantum state. In other words, a large advantage can be gained by using quantum memory. Since fermionic Majorana operators can be mapped to qubit Pauli operators using the Jordan-Wigner transformation, known methods for learning Pauli operations can be used to learn expectation values of fermionic Majorana operators (which can be used to determine the k-RDMs that describe the properties of the quantum state). These known methods have asample complexity of ^^^log ^^ / ^^ସ^ and require entangled measurements on Ω^1 / ^^ଶ^ copiesof the quantum state to determine the signs of the expectation values (where ^^ represents a target precision). This means that on Ω^1 / ^^ଶ^ copies of the quantum state need to be held in quantum memory simultaneously to perform the required entangled measurements. Such large-scale entanglement requires significant quantum memory, which may not be feasible for some quantum computing architectures, e.g., early fault tolerant quantum devices. The presently described techniques improve on these know methods. For example, examples of the presently described techniques can be used to estimate Majorana operator expectation values to a target precision with a reduced quantum memory requirement. In particular, the techniques include measurement protocols that enable efficient estimation of M expectation values of Majorana operators to precision ^^ (with their signs) using^^^log ^^ / ^^ଶ^ single-copy and two-copy measurements. By restricting the entangledmeasurements to at most two-copies of the quantum state at a time (i.e., measuring ^^ ⊗ ^^instead of a tensor product of Ω^1 / ^^ଶ^ copies of the quantum state ^^), the quantum memory Attorney Docket: 56113-0625WO1 requirement is reduced since the quantum computer only needs to store and process at most two copies of the quantum state at once (instead of Ω^1 / ^^ଶ^ copies of the quantum state). This in turn allows for the above described exponential advantages to be implemented in the early fault-tolerant regime. Further, by reducing the required number of simultaneously held copies of the quantum state, the techniques can be implemented using less qubits (since holding Ω^1 / ^^ଶ^ copies of the quantum state requires more qubits in memory than holding two copies of the quantum state). The techniques are therefore particularly suitable for quantum devices with a limited number of qubits, e.g., near term quantum devices. Details of one or more implementations of the subject matter of this specification are set forth in the accompanying drawings and the description below. Other features, aspects, and advantages of the subject matter will become apparent from the description, the drawings, and the claims. BRIEF DESCRIPTION OF THE DRAWINGS FIG.1 depicts an example system for measuring fermionic reduced density matrices. FIG.2 is a flow diagram of an example process for measuring quadratic Majorana observables. FIG.3 is a flow diagram of an example process for measuring k-body Majorana observables FIG.4 is a flow diagram of an example process for measuring expectation values of quadratic Pauli operators up to a sign. FIG.5 is a flow diagram of an example measurement protocol for measuring quadratic Majorana observables. FIG.6 is a flow diagram of an example measurement protocol for measuring k-body Majorana observables. FIG.7 shows an example quantum computing device. DETAILED DESCRIPTION FIG.1 is a block diagram of an example system 100 for measuring elements of fermionic operators, e.g., k-body Majorana operators. The example system 100 is an example of a system implemented as classical and quantum computer programs on one or more classical computers and quantum computing devices in one or more locations, in which the systems, components, and techniques described herein can be implemented. Attorney Docket: 56113-0625WO1 The example system 100 includes a quantum computing device 102. The quantum computing device 102 includes classical and quantum computing components. Components of the quantum computing device 102 can be through a network, e.g., a local area network (LAN), wide area network (WAN), the Internet, or a combination thereof. The classical components, e.g., the graph generator 106 and a basis generator 110, are configured to perform classical computations and can be implemented as one or more computer programs, i.e., one or more modules of computer program instructions encoded on a tangible non-transitory storage medium for execution by, or to control the operation of, a data processing apparatus. The computer storage medium can be a machine-readable storage device, a machine-readable storage substrate, a random or serial access memory device, or a combination of one or more of them. The quantum computing hardware is configured to perform quantum computation. For example, the quantum computing device 102 can include a qubit array 122, quantum memory 120, quantum circuitry 114, and control devices configured to operate physical qubits in the qubit array and apply quantum circuits to the qubits. In some implementations the quantum computing device 102 can be a noisy device, e.g., a noisy intermediate-scale quantum device (NISQ device), a trapped ion quantum computer, or a neutral atom quantum computer. An example quantum computing device is described in the more detail below with reference to FIG.7. The quantum computing device 102 is configured to receive an input 104 that includes a list of ^^ k-body Majorana operators and a predefined precision. The Majorana operators can be defined as a set of mutually anticommuting Hermitian observables ^^^with ^^ற^ ൌ ^^^ , ^^^^ , ^^^^ ൌ 2^^^^ where ^^ represents the Kronecker delta. There are two Majoranaoperators associated to every fermionic mode, with creation and annihilation operators ^^^,^^ற^.They are related by ^^ଶ^ି^ ൌ ^^^ ^ ^^ற^ , ^^ଶ^ ൌ െ^^൫^^^ െ ^^ற^൯and ^^^ൌ ^ ଶ൫^^ଶ^ି^ ^ ^^^^ଶ^൯, ^^ற^ ൌ ^ ଶ൫^^ െ ^^^^ଶ^ to operators on qubits via ^^ଶ^ି^ ൌ ^^^ ⊗ … ⊗ ^^^ ⊗ ^^^ ⊗ ^^⊗ …⊗ ^^^^ଶ^ ൌ ^^^ ⊗ … ⊗ ^^^ ⊗ ^^^ ⊗ ^^⊗ …⊗ ^^ Attorney Docket: 56113-0625WO1where ^^^,^^^, and ^^^ represent Pauli-X, Y, and Z operators (where the ^^^ and ^^^ act on qubitj, and the ^^^act on the first j-1 qubits) and ^^ is an identity operator (which acts on the last P-j qubits in a set of P qubits). A k-body Majorana operator is a Hermitian product of 2kMajorana operators. For example, χ ൌ ^^^^^^^ଶ is a 1-body Majorana operator. 1-bodyMajorana operators are also referred to herein as quadratic Majorana operators. The graph generator 106 is configured to obtain a list of expectation values for the ^^ k-body Majorana operators, where each expectation value in the list corresponds to a respective k-body Majorana operator. The graph generator 106 can receive the list of expectation values, e.g., as input from an external party, or performing classical and quantum computing operations to generate the list of expectation values. Example operations performed by the graph generator 106 to generate the list of expectation values is described below with reference to FIG.4. The graph generator 106 is then configured to construct a graph for the ^^ k-body Majorana operators using the list of expectation values. That is, the graph generator 106 can generate one or more data structures that represent a graph for the the ^^ k-body Majorana operators, e.g., including an adjacency matrix, adjacency list and adjacency set. The graph generator 106 is configured to implement a coloring algorithm to color the graph. In some implementations the coloring algorithm is an edge coloring algorithm, in other implementations the coloring algorithm is a vertex coloring algorithm. Example graphs and coloring algorithm are described in more detail below with reference to FIGS.2 and 3. The basis generator 110 is configured to use the colored graph to determine entangled measurements to perform to compute the expectation values of the Majorana operators. For example, the basis generator 110 can use the colored graph to determine simultaneous eigenbases of k-body Majorana operators in the graph (or represented by respective edges and vertices in the graph) with a same color. The basis generator 110 can provide the determined measurement bases to the quantum computing hardware. The quantum computing hardware is configured to perform measurements of the k- body Majorana operators on copies of a quantum state stored in the quantum memory 120 using the determined bases. Measured values of the k-body Majorana operators are provided as output, e.g., as part of a quantum chemistry simulation. FIG.2 is a flow diagram of an example process 200 for measuring quadratic Majorana observables. For convenience, the process 200 will be described as being Attorney Docket: 56113-0625WO1 performed by quantum hardware in communication with control electronics located in one or more locations. For example, the system 100 of FIG.1, appropriately programmed in accordance with this specification, can perform the process 200. The system obtains an input that includes a list of ^^ quadratic Majorana operators ^^^, … ,^^ெ where each quadratic Majorana operator is equal to a product of respectiveMajorana operators multiplied by ^^, that is ^^^ ൌ ^^^^^ೕ^^^ೕ where ^^ ∈ ^1, … , ^^^ and ^^^ , ^^^ areindices for the corresponding fermion modes. The input also includes a predefined precision ^^ and ^^ copies of a quantum state ^^ ⊗ ^^ (step 202). The quantum state ^^⊗ ^^ is a combinedquantum state of two independent copies of an unknown quantum state ^^ on ^^ fermion modes, i.e., a tensor product of the quantum state ^^ on ^^ fermion modes with itself. The ^^ copies of the quantum state can be stored and retrieved from a quantum memory. The system processes the input to obtain a list of expectation values ^^^^^, … ,^^^ெ^,where each expectation value in the list corresponds to a respective quadratic Majorana operator (step 204). An example process that can be used at step 204 to process the input is described in more detail below with reference to FIG.4. By construction, the system requires ^^^ ൌ ^^ ^^^^ெ ఢర^ copies of the tensor product ^^⊗ ^^ to process the input and obtain the list ofwhere ^^ represents the predefined precision. The system constructs a graph (e.g., generates data structures that represent a graph) (step 206). The system constructs the graph by assigning single Majorana modes included in the ^^ quadratic Majorana operators to respective vertices in the graph. That is, vertices in the graph are the single Majorana modes ^^^^, … , ^^ଶ^^. The system then adds edges betweenvertices that represent a pair of single Majorana modes that correspond to a non-zero expectation value in the list. Since quadratic Majorana operators are defined as products of two single Majorana modes, edges of the graph naturally represent respective quadratic Majorana operators. That is, for each ^^ such that ^^^^ ് 0, the graph includes an edge^^^^ೕ , ^^^ೕ^. system applies an edge coloring algorithm to color the graph (e.g., label data in the data structures) in multiple different colors, where edges incident on a vertex are assigned different colors (step 208). Two distinct quadratic Majorana observables commute if and only if they do not share any Majorana modes. For example, ^^^^^^^ଶanti-commutes with ^^^^ଶ^^ଷbut commutes with ^^^^ଷ^^ସ. Thus, a commuting group of quadratic Majorana operators corresponds to a matching in the graph, and partitioning a set of quadratic Majorana operators Attorney Docket: 56113-0625WO1 into commuting groups corresponds to an edge coloring. In some implementations, the number of colors used to color the graph is at most deg(G) + 1, where deg(G) represents a maximum degree of any vertex in the graph. The system measures each quadratic Majorana operator in the list of ^^ quadratic Majorana operators using entangled measurements that are based on the colors assigned to the edges in the graph (step 210). For each quadratic Majorana operator, the system identifies a color assigned to an edge that represents the quadratic Majorana operator and measures multiple copies of the quantum state ^^ in a simultaneous eigenbasis of all quadratic Majorana operators that are represented by the color in the graph to obtain an empirical estimate of the quadratic Majorana operator. For example, if edges that correspond to quadratic Majorana operators ^^^,^^ଶ,^^^are colored in a same first color and edges that correspond to quadratic Majorana operators ^^ଷ,^^ெି^,^^ெare colored in a same second color that is different to the first color, then quadratic Majorana operator ^^^is measured in asimultaneous eigenbasis of ^^^,^^ଶ, and ^^^, quadratic Majorana operator ^^ଶ is measured in asimultaneous eigenbasis of ^^^,^^ଶ, and ^^^, quadratic Majorana operator ^^^ is measured in asimultaneous eigenbasis of ^^^,^^ଶ, and ^^^, quadratic Majorana operator ^^ଷ is measured in asimultaneous eigenbasis of ^^ଷ,^^ெି^, and ^^ெ, quadratic Majorana operator ^^ெି^ ismeasured in a simultaneous eigenbasis of ^^ଷ,^^ெି^, and ^^ெ, and quadratic Majoranaoperator ^^ெ is measured in a simultaneous eigenbasis of ^^ଷ,^^ெି^, and ^^ெ.Each empirical estimate ^^^^of a respective quadratic Majorana operator ^^^estimates a trace of the quadratic Majorana operator ^^^with respect to the quantum state ^^. By construction, a magnitude of a difference between each empirical estimate of a quadratic Majorana operator and a trace of the quadratic Majorana operator with respect to the quantumstate is less than or equal to the predefined precision, that is ห^^^^ െ Tr^^^^^^^ห ^ ^^.The system requires ^^୪୭^ெ ଶൌ ^^ ^ఢమ^ copies of the quantum state to obtain an empirical estimate of each quadratic in the list of ^^ quadratic Majorana operators. The total sample complexity ^^ of example process 200 is therefore ^^^^^^^^ 1 ^^^^^^^^ ^^^^^^^^ ^^^ ^ ^ ^^ ^ ^ ^ 1^^^ ^ ^^ ^ since it can be shown that deg^G^ ^^ Attorney Docket: 56113-0625WO1 In some implementations the system can convert one or more of the empirical estimates of the quadratic Majorana operators to respective estimates of one or more creation or annihilation operators. For example, given estimates ^^^^ ^^^^ for all k-body Majoranas ^^, the equations ^^ ^ற ^^ൌ ଶ൫^^ଶ^ି^ ^ ^^^^ଶ^൯ and ^^^ ൌଶ൫^^ଶ^ି^ െ ^^^^ଶ^൯ can be used to write ^^^^ ^^^^ asa sum over ^^^^^ ^^^^^ estimates of ^^^^ ^^^^ are correct to precision ^^, the of ^^^^ ^^^^ are also correct to precision ^^. FIG.3 is a flow diagram of an example process 300 for measuring k-body Majorana observables. For convenience, the process 300 will be described as being performed by quantum hardware in communication with control electronics located in one or more locations. For example, the system 100 of FIG.1, appropriately programmed in accordance with this specification, can perform the process 300. The system obtains an input that includes a list of ^^ k-body Majorana operators^^^, … ,^^ெ where each quadratic Majorana operator is equal to a respective Hermitian productof 2^^ Majorana operators multiplied by ^^. The input also includes a predefined precision ^^and ^^ copies of a quantum state of ^^ ⊗ ^^, where the ^^ copies of the quantum state can bestored and retrieved from a quantum memory (step 302). The system processes the input to obtain a list of expectation values ^^^^^, … ,^^^ெ^,where each expectation value in the list corresponds to a respective k-body Majorana operator (step 304). Step 304 of example process 300 is similar to step 202 of example process 200, and for brevity, details are not repeated. The system constructs a graph (e.g., generates data structures that represent a graph) (step 306). For each non-zero expectation value in the list, the system adds a corresponding vertex to the graph. Since each expectation value corresponds to a respective k-body Majorana operator, each vertex in the graph also corresponds to a respective quadratic Majorana operator. The system adds edges between vertices in the graph that representanticommuting quadratic Majorana operators, e.g., adds an edge ^^^, ^^^ for each ^^,^^ where ^^^and ^^^anti-commute. The system applies a vertex coloring algorithm to color the graph (i.e., label data in the data structures) in multiple colors, where adjacent vertices have different colors and a total number of colors used is minimized (step 308). In some implementations, the total Attorney Docket: 56113-0625WO1 number of colors used to color the graph is less than or equal to a number of an order of ^ ఢమwhere ^^ represents the predefined precision. The system measures each quadratic Majorana operator in the list of ^^ Majorana operators using entangled measurements that are based on the colors assigned to the graph vertices (step 310). For each quadratic Majorana operator, the system identifies a color assigned to a vertex that represents the quadratic Majorana operator, and measures multiple copies of the quantum state ^^ in a simultaneous eigenbasis of all quadratic Majorana operators that are represented by vertices with the color to obtain an empirical estimate of the quadratic Majorana operator. Step 310 is similar to step 210 of example process 200, and further details are not repeated. Each empirical estimate ^^^^of a respective quadratic Majorana operator ^^^estimates a trace of the quadratic Majorana operator ^^^with respect to the quantum state ^^. By construction, a magnitude of a difference between each empirical estimate of a quadratic Majorana operator and a trace of the quadratic Majorana operator with respect to the quantumstate is less than or equal to the predefined precision, that is ห^^^^ െ Tr^^^^^^^ห ^ ^^.The system requires ^^ ൌ ^^୪୭^ெ ଶ^ఢమ^ copies of the quantum state to obtain an empirical estimate of each k-body in the list of ^^ k-body Majorana operators. The total sample complexity ^^ of example process 200 is therefore ^^ ^ ^^^^ ൌ^^^^^^^^ ^^^^^^^^^^ ^ଶ ^^ ൬ ^ ^ ^^ ൬^ ^ where ^^ represents the number of colors used at step 308. It can be shown that the number ofcolors required to color the graph does not exceed ^^ ^^ ఢమ^. As described above with reference to 200, in some implementations the system can convert one or more of the empirical estimates of the quadratic Majorana operators to respective estimates of one or more creation or annihilation operators. FIG.4 is a flow diagram of an example process 400 for measuring expectation values of Pauli operators up to a sign. In particular, example process 400 can be used to output estimates of the magnitude of the expectation value ^^^^^^^^^^^for each Pauli operator ^^^in a Attorney Docket: 56113-0625WO1set of Pauli operators ^^^, … ,^^ெ on n qubits with predefined precision ^^ using multiple copiesof the quantum state ^^. For convenience, 400 will be described as being performed by quantum hardware in communication with control electronics located in one or more locations. For example, the system 100 of FIG.1, appropriately programmed in accordance with this specification, can perform the process 400. Example process 400 can be used in combination with example processes 200 and 300 described above, for example at step 204 and step 304 of example process 200 and 300, respectively, to process the input received at steps 202 and 302. The system measures each copy of ^^ copies of a tensor product ^^⊗ ^^ in an ^^-foldBell basis ^|Φ^^⃗ ,^^⃗ ^ ∶ ൫^⃗^,^^^⃗ ൯ ∈ ^0,1^ଶ^^to obtain a set of measurement outcomes^൫^⃗^^^^, ^^^⃗ ^^^ . Each measurement outcome in the set corresponds to a respective measurement of the tensor product and includes a first binary string ^⃗^^^^and asecond binary string ^^^⃗ ^^^. Step 402 therefore converts the N copies of the quantum state ^^⊗^^ to a classical dataset. From this dataset, the magnitude of the expectation value |^^^^^^^^^^^| for Pauli operator ^^^can be estimated, as described below with reference to steps 404-408b. The ^^ copies of the tensor product ^^⊗ ^^ can be stored simultaneously in a quantummemory during the measurement process. Since the measurements performed at step 402 are entangled measurements that are restricted to only two copies of the quantum state ^^ at a time(i.e., the tensor product ^^ ⊗ ^^ is measured), the quantum memory requirement is thereforereduced, e.g., compared to conventional Pauli learning algorithms that require entangled measurements on Ω^ ^ ఢమ^ copies of the quantum state at a time. The the following steps for each of the ^^ quadratic Majoranaoperators ^^^, … ,^^ெ (i.e., their Pauli operator representations). For a given quadraticMajorana operator that is mapped to a corresponding quadratic Pauli operator, the system computes a sum of values of a variable, where each value in the sum corresponds to a respective measurement outcome obtained at step 402. The value of the variable is equal to 1 if a product of a first binary string included in a respective measurement outcome and a binary representation of a first Pauli operator included in the quadratic Pauli operator minus a product of a second binary string included in the respective measurement outcome and a binary representation of a second Pauli operator included in the quadratic Pauli operator is equal to 0 modulo 2 or is equal to -1 if the first binary string included in the respective Attorney Docket: 56113-0625WO1 measurement outcome multiplied by the binary representation of the first Pauli operator included in the quadratic Pauli operator minus the second binary string included in the respective measurement outcome multiplied by the binary representation of the second Pauli operator included in the quadratic Pauli operator is equal to 1 modulo 2 (step 404). That is, for the quadratic Majorana operator ^^^ ൌ ^^ଶ^ି^^^ଶ^ that maps to the qubit operator ^^^ ൌ^^^^⃗ ೕ,^ೕ⃗ , where ^^^^⃗ ೕ,^ೕ⃗ ൌ ^^^భ,^భ ⊗ …⊗ ^^^^,^^ with ^^^,^ ൌ ^^, ^^^,^ ൌ ^^^, ^^^,^ ൌ ^^^, and ^^^,^ ൌ^^ 1if ^⃗^^^ ^⃗^^^^ ^^ ^ ^ ∙ ^^^ െ ^^ ∙ ^⃗^^ ൌ 0 mod 2^^^ൌ ே∑ே^^^^^ୀ^ ^^^where ^^^ ൌ ^^⃗^^^^ ∙ ^^ ^^^⃗^^^ ∙ ^⃗^^ 1 mod to a first predetermined threshold that depends on the predefined precision (step 406). The first predetermined threshold is equal to 2 / 3 of a square of the predefined precision. That is, the system determines whether ^^^^^ ଶ ଷ ^^ଶ. In response to determining that the sum of values is less than or equal to the first threshold, the system outputs the value ^^^^ ൌ 0 (step 408a). Outputting the value 0 indicates that a magnitude of a trace of the corresponding quadratic Pauli operator ^^^with respect to the quantum state ^^ on n fermion modes is less than or equal to the predefined precision ^^, i.e., indicates that ห^^^^^^^^^^^ห ^ ^^.Alternatively, in response to determining that the sum of values is greater than the first predetermined threshold, the system outputs a square root of the sum of values (step 408b). That is, the system outputs ^^^^ ൌ ^^^^^. Outputting the square root of the sum of valuesindicates that a magnitude of a difference between a positive or negative square root of the sum of values and a trace of the corresponding quadratic Pauli operator ^^^with respect to the quantum state ^^ on n fermion modes is less than or equal to a second predetermined threshold that is based on the predefined precision ^^, i.e., indicates that หേ^^^√ଶ ^െ ^^^^^^^^^^^ห ^^^ for one of the choices of sign േ, where√ଶ ଶ√ଷ ^^ represents the FIG.5 is a flow diagram of an example measurement protocol 500 for measuring quadratic Majorana observables. For convenience, the process 500 will be described as being performed by quantum hardware in communication with a control and measurement system Attorney Docket: 56113-0625WO1 located in one or more locations. For example, the system 100 of FIG.1, appropriately programmed in accordance with this specification, can perform the process 500. Example process 500 can be combined with the techniques described above with reference to example process 200 of FIG.2. The system obtains an input that includes a list of M quadratic Majorana operators, a predefined precision, and N copies of a tensor product of a quantum state on n fermion modes and the quantum state on the n fermion modes (step 502). The system determines a basis to measure each quadratic Majorana operator in the list in (step 504). To determine the basis for a specific quadratic Majorana operator in the list, the system processes the input to obtain a list of expectation values, where each expectation value in the list corresponds to a respective quadratic Majorana operator (step 506a). The system constructs a graph by assigning single Majorana modes included in the M quadratic Majorana operators to respective vertices in the graph and adding edges between vertices that represent a pair of single Majorana modes that correspond to a non-zero expectation value in the list (step 506b). The system applies an edge coloring algorithm to color the graph in multiple colors, where incident edges on a vertex have different colors (step 506c). The system determines a simultaneous eigenbasis of quadratic Majorana operators in the graph that have a same color that corresponds to the quadratic Majorana operator (step 506d). The system measures the quadratic Majorana operator in the determined basis to obtain an empirical estimate of the quadratic Majorana operator (step 508). For example, the system can compile a quantum circuit that implements a measurement in the determined basis and generate control signals that apply the quantum circuit to a qubit system that corresponds to the fermion system (e.g., a qubit system obtained through application of the Jordan-Wigner transformation). In some implementations the quantum circuit that implements the measurement in the determined basis can include a sequence of single qubit rotations that precede a Z-measurement or measurement in the computational basis. FIG.6 is a flow diagram of an example measurement protocol 600 for measuring k- body Majorana observables. For convenience, the process 600 will be described as being performed by quantum hardware in communication with a control and measurement system located in one or more locations. For example, the system 100 of FIG.1, appropriately programmed in accordance with this specification, can perform the process 600. Example process 600 can be combined with the techniques described above with reference to example process 300 of FIG.3. Attorney Docket: 56113-0625WO1 The system obtains an input that includes a list of ^^ k-body Majorana operators, a predefined precision, and ^^ copies of a tensor product of a quantum state on ^^ fermion modes and the quantum state on the ^^ fermion modes (step 602). The system determines a basis to measure each k-body Majorana operator in the list in (step 604). To determine the basis for a specific quadratic Majorana operator in the list, the system processes the input to obtain a list of expectation values, where each expectation value in the list corresponds to a respective quadratic Majorana operator (step 606a). The system constructs a graph by, for each non-zero expectation value in the list, adding a vertex to the graph that represents a k-body Majorana operator that corresponds to the non-zero expectation value and adding edges between vertices in the graph that represent anticommuting k-body Majorana operators (step 606b). The system applies a vertex coloring algorithm to color the graph in multiple colors, where adjacent vertices have different colors and a total number of colors used is minimized (step 606c). The system determines a simultaneous eigenbasis of k-body Majorana operators in the graph that have a same color that corresponds to the k-body Majorana operator (step 606d). The system measures the k-body Majorana operator in the determined basis to obtain an empirical estimate of the k-body Majorana operator (step 608). For example, the system can compile a quantum circuit that implements a measurement in the determined basis and generate control signals that apply the quantum circuit to a qubit system that corresponds to the fermion system (e.g., a qubit system obtained through application of the Jordan-Wigner transformation). In some implementations the quantum circuit that implements the measurement in the determined basis can include a sequence of single qubit rotations that precede a Z-measurement or measurement in the computational basis. FIG.7 depicts an example quantum computer 700 for performing the quantum operations described in this specification. The example quantum computer 700 includes an example quantum computing device 702. The quantum computing device 702 is intended to represent various forms of quantum computing devices. The components shown here, their connections and relationships, and their functions, are exemplary only, and do not limit implementations of the inventions described and / or claimed in this document. The example quantum computing device 702 includes a qubit assembly 752 and a control and measurement system 704. The qubit assembly includes multiple qubits, e.g., qubit 706, that are used to perform algorithmic operations or quantum computations. While Attorney Docket: 56113-0625WO1 the qubits shown in FIG.7 are arranged in a rectangular array, this is a schematic depiction and is not intended to be limiting. The qubit assembly 752 also includes adjustable coupling elements, e.g., coupler 708, that allow for interactions between coupled qubits. In the schematic depiction of FIG.7, each qubit is adjustably coupled to each of its four adjacent qubits by means of respective coupling elements. However, this is an example arrangement of qubits and couplers and other arrangements are possible, including arrangements that are non-rectangular, arrangements that allow for coupling between non-adjacent qubits, and arrangements that include adjustable coupling between more than two qubits. Each qubit can be a physical two-level quantum system or device having levels representing logical values of 0 and 1. The specific physical realization of the multiple qubits and how they interact with one another is dependent on a variety of factors including the type of the quantum computing device 702 included in the example computer 700 or the type of quantum computations that the quantum computing device is performing. For example, in an atomic quantum computer the qubits may be realized via atomic, molecular or solid-state quantum systems, e.g., hyperfine atomic states. As another example, in a superconducting quantum computer the qubits may be realized via superconducting qubits or semi-conducting qubits, e.g., superconducting transmon states. As another example, in a NMR quantum computer the qubits may be realized via nuclear spin states. In some implementations a quantum computation can proceed by loading qubits, e.g., from a quantum memory, and applying a sequence of unitary operators to the qubits. Applying a unitary operator to the qubits can include applying a corresponding sequence of quantum logic gates to the qubits, e.g., to implement the surface code circuits described in this specification. Example quantum logic gates include single-qubit gates, e.g., Pauli-X, Pauli-Y, Pauli-Z (also referred to as X, Y, Z), Hadamard gates, S gates, rotations, two-qubit gates, e.g., controlled-X, controlled-Y, controlled-Z (also referred to as CX, CY, CZ), controlled NOT gates (also referred to as CNOT) controlled swap gates (also referred to as CSWAP), iSWAP gates, and gates involving three or more qubits, e.g., Toffoli gates. The quantum logic gates can be implemented by applying control signals 710 generated by the control and measurement system 704 to the qubits and to the couplers. For example, in some implementations the qubits in the qubit assembly 752 can be frequency tunable. In these examples, each qubit can have associated operating frequencies that can be adjusted through application of voltage pulses via one or more drive-lines coupled to the qubit. Example operating frequencies include qubit idling frequencies, qubit interaction frequencies, and qubit readout frequencies. Different frequencies correspond to Attorney Docket: 56113-0625WO1 different operations that the qubit can perform. For example, setting the operating frequency to a corresponding idling frequency may put the qubit into a state where it does not strongly interact with other qubits, and where it may be used to perform single-qubit gates. As another example, in cases where qubits interact via couplers with fixed coupling, qubits can be configured to interact with one another by setting their respective operating frequencies at some gate-dependent frequency detuning from their common interaction frequency. In other cases, e.g., when the qubits interact via tunable couplers, qubits can be configured to interact with one another by setting the parameters of their respective couplers to enable interactions between the qubits and then by setting the qubit’s respective operating frequencies at some gate-dependent frequency detuning from their common interaction frequency. Such interactions may be performed in order to perform multi-qubit gates. The type of control signals 710 used depends on the physical realizations of the qubits. For example, the control signals may include RF or microwave pulses in an NMR or superconducting quantum computer system, or optical pulses in an atomic quantum computer system. A quantum computation can be completed by measuring the states of the qubits, e.g., using a quantum observable such as X or Z, using respective control signals 710. The measurements cause readout signals 712 representing measurement results to be communicated back to the measurement and control system 704. The readout signals 712 may include RF, microwave, or optical signals depending on the physical scheme for the quantum computing device and / or the qubits. For convenience, the control signals 710 and readout signals 712 shown in FIG.7 are depicted as addressing only selected elements of the qubit assembly (i.e. the top and bottom rows), but during operation the control signals 710 and readout signals 712 can address each element in the qubit assembly 752. The control and measurement system 704 is an example of a classical computer system that can be used to perform various operations on the qubit assembly 752, as described above, as well as other classical subroutines or computations. The control and measurement system 704 includes one or more classical processors, e.g., classical processor 714, one or more memories, e.g., memory 716, and one or more I / O units, e.g., I / O unit 718, connected by one or more data buses. The control and measurement system 704 can be programmed to send sequences of control signals 710 to the qubit assembly, e.g. to carry out a selected series of quantum gate operations, and to receive sequences of readout signals 712 from the qubit assembly, e.g. as part of performing measurement operations. Attorney Docket: 56113-0625WO1 The processor 714 is configured to process instructions for execution within the control and measurement system 704. In some implementations, the processor 714 is a single-threaded processor. In other implementations, the processor 714 is a multi-threaded processor. The processor 714 is capable of processing instructions stored in the memory 716. The memory 716 stores information within the control and measurement system 704. In some implementations, the memory 716 includes a computer-readable medium, a volatile memory unit, and / or a non-volatile memory unit. In some cases, the memory 716 can include storage devices capable of providing mass storage for the system 704, e.g. a hard disk device, an optical disk device, a storage device that is shared over a network by multiple computing devices (e.g., a cloud storage device), and / or some other large capacity storage device. The input / output device 718 provides input / output operations for the control and measurement system 704. The input / output device 718 can include D / A converters, A / D converters, and RF / microwave / optical signal generators, transmitters, and receivers, whereby to send control signals 710 to and receive readout signals 712 from the qubit assembly, as appropriate for the physical scheme for the quantum computer. In some implementations, the input / output device 718 can also include one or more network interface devices, e.g., an Ethernet card, a serial communication device, e.g., an RS-232 port, and / or a wireless interface device, e.g., an 802.11 card. In some implementations, the input / output device 718 can include driver devices configured to receive input data and send output data to other external devices, e.g., keyboard, printer and display devices. Although an example control and measurement system 704 has been depicted in FIG. 7, implementations of the subject matter and the functional operations described in this specification can be implemented in other types of digital electronic circuitry, or in computer software, firmware, or hardware, including the structures disclosed in this specification and their structural equivalents, or in combinations of one or more of them. Implementations of the digital and / or quantum subject matter and the digital functional operations and quantum operations described in this specification can be implemented in digital electronic circuitry, suitable quantum circuitry or, more generally, quantum computational systems, in tangibly-embodied digital and / or quantum computer software or firmware, in digital and / or quantum computer hardware, including the structures disclosed in this specification and their structural equivalents, or in combinations of one or more of them. The term “quantum computational systems” may include, but is not limited to, quantum computers, quantum information processing systems, quantum cryptography systems, or quantum simulators. Attorney Docket: 56113-0625WO1 Implementations of the digital and / or quantum subject matter described in this specification can be implemented as one or more digital and / or quantum computer programs, i.e., one or more modules of digital and / or quantum computer program instructions encoded on a tangible non-transitory storage medium for execution by, or to control the operation of, data processing apparatus. The digital and / or quantum computer storage medium can be a machine-readable storage device, a machine-readable storage substrate, a random or serial access memory device, one or more qubits, or a combination of one or more of them. Alternatively or in addition, the program instructions can be encoded on an artificially- generated propagated signal that is capable of encoding digital and / or quantum information, e.g., a machine-generated electrical, optical, or electromagnetic signal, that is generated to encode digital and / or quantum information for transmission to suitable receiver apparatus for execution by a data processing apparatus. The terms quantum information and quantum data refer to information or data that is carried by, held or stored in quantum systems, where the smallest non-trivial system is a qubit, i.e., a system that defines the unit of quantum information. It is understood that the term “qubit” encompasses all quantum systems that may be suitably approximated as a two- level system in the corresponding context. Such quantum systems may include multi-level systems, e.g., with two or more levels. By way of example, such systems can include atoms, electrons, photons, ions or superconducting qubits. In many implementations the computational basis states are identified with the ground and first excited states, however it is understood that other setups where the computational states are identified with higher level excited states are possible. The term “data processing apparatus” refers to digital and / or quantum data processing hardware and encompasses all kinds of apparatus, devices, and machines for processing digital and / or quantum data, including by way of example a programmable digital processor, a programmable quantum processor, a digital computer, a quantum computer, multiple digital and quantum processors or computers, and combinations thereof. The apparatus can also be, or further include, special purpose logic circuitry, e.g., an FPGA (field programmable gate array), an ASIC (application-specific integrated circuit), or a quantum simulator, i.e., a quantum data processing apparatus that is designed to simulate or produce information about a specific quantum system. In particular, a quantum simulator is a special purpose quantum computer that does not have the capability to perform universal quantum computation. The apparatus can optionally include, in addition to hardware, code that creates an execution environment for digital and / or quantum computer programs, e.g., code that constitutes Attorney Docket: 56113-0625WO1 processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of them. A digital computer program, which may also be referred to or described as a program, software, a software application, a module, a software module, a script, or code, can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and it can be deployed in any form, including as a stand-alone program or as a module, component, subroutine, or other unit suitable for use in a digital computing environment. A quantum computer program, which may also be referred to or described as a program, software, a software application, a module, a software module, a script, or code, can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and translated into a suitable quantum programming language, or can be written in a quantum programming language, e.g., QCL or Quipper. A digital and / or quantum computer program may, but need not, correspond to a file in a file system. A program can be stored in a portion of a file that holds other programs or data, e.g., one or more scripts stored in a markup language document, in a single file dedicated to the program in question, or in multiple coordinated files, e.g., files that store one or more modules, sub-programs, or portions of code. A digital and / or quantum computer program can be deployed to be executed on one digital or one quantum computer or on multiple digital and / or quantum computers that are located at one site or distributed across multiple sites and interconnected by a digital and / or quantum data communication network. A quantum data communication network is understood to be a network that may transmit quantum data using quantum systems, e.g. qubits. Generally, a digital data communication network cannot transmit quantum data, however a quantum data communication network may transmit both quantum data and digital data. The processes and logic flows described in this specification can be performed by one or more programmable digital and / or quantum computers, operating with one or more digital and / or quantum processors, as appropriate, executing one or more digital and / or quantum computer programs to perform functions by operating on input digital and quantum data and generating output. The processes and logic flows can also be performed by, and apparatus can also be implemented as, special purpose logic circuitry, e.g., an FPGA or an ASIC, or a quantum simulator, or by a combination of special purpose logic circuitry or quantum simulators and one or more programmed digital and / or quantum computers. Attorney Docket: 56113-0625WO1 For a system of one or more digital and / or quantum computers to be “configured to” perform particular operations or actions means that the system has installed on it software, firmware, hardware, or a combination of them that in operation cause the system to perform the operations or actions. For one or more digital and / or quantum computer programs to be configured to perform particular operations or actions means that the one or more programs include instructions that, when executed by digital and / or quantum data processing apparatus, cause the apparatus to perform the operations or actions. A quantum computer may receive instructions from a digital computer that, when executed by the quantum computing apparatus, cause the apparatus to perform the operations or actions. Digital and / or quantum computers suitable for the execution of a digital and / or quantum computer program can be based on general or special purpose digital and / or quantum processors or both, or any other kind of central digital and / or quantum processing unit. Generally, a central digital and / or quantum processing unit will receive instructions and digital and / or quantum data from a read-only memory, a random access memory, or quantum systems suitable for transmitting quantum data, e.g. photons, or combinations thereof. The essential elements of a digital and / or quantum computer are a central processing unit for performing or executing instructions and one or more memory devices for storing instructions and digital and / or quantum data. The central processing unit and the memory can be supplemented by, or incorporated in, special purpose logic circuitry or quantum simulators. Generally, a digital and / or quantum computer will also include, or be operatively coupled to receive digital and / or quantum data from or transfer digital and / or quantum data to, or both, one or more mass storage devices for storing digital and / or quantum data, e.g., magnetic, magneto-optical disks, optical disks, or quantum systems suitable for storing quantum information. However, a digital and / or quantum computer need not have such devices. Digital and / or quantum computer-readable media suitable for storing digital and / or quantum computer program instructions and digital and / or quantum data include all forms of non-volatile digital and / or quantum memory, media and memory devices, including by way of example semiconductor memory devices, e.g., EPROM, EEPROM, and flash memory devices; magnetic disks, e.g., internal hard disks or removable disks; magneto-optical disks; CD-ROM and DVD-ROM disks; and quantum systems, e.g., trapped atoms or electrons. It is understood that quantum memories are devices that can store quantum data for a long time with high fidelity and efficiency, e.g., light-matter interfaces where light is used for Attorney Docket: 56113-0625WO1 transmission and matter for storing and preserving the quantum features of quantum data such as superposition or quantum coherence. Control of the various systems described in this specification, or portions of them, can be implemented in a digital and / or quantum computer program product that includes instructions that are stored on one or more non-transitory machine-readable storage media, and that are executable on one or more digital and / or quantum processing devices. The systems described in this specification, or portions of them, can each be implemented as an apparatus, method, or system that may include one or more digital and / or quantum processing devices and memory to store executable instructions to perform the operations described in this specification. While this specification contains many specific implementation details, these should not be construed as limitations on the scope of what may be claimed, but rather as descriptions of features that may be specific to particular implementations. Certain features that are described in this specification in the context of separate implementations can also be implemented in combination in a single implementation. Conversely, various features that are described in the context of a single implementation can also be implemented in multiple implementations separately or in any suitable sub-combination. Moreover, although features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be excised from the combination, and the claimed combination may be directed to a sub-combination or variation of a sub-combination. Similarly, while operations are depicted in the drawings in a particular order, this should not be understood as requiring that such operations be performed in the particular order shown or in sequential order, or that all illustrated operations be performed, to achieve desirable results. In certain circumstances, multitasking and parallel processing may be advantageous. Moreover, the separation of various system modules and components in the implementations described above should not be understood as requiring such separation in all implementations, and it should be understood that the described program components and systems can generally be integrated together in a single software product or packaged into multiple software products. Particular implementations of the subject matter have been described. Other implementations are within the scope of the following claims. For example, the actions recited in the claims can be performed in a different order and still achieve desirable results. As one example, the processes depicted in the accompanying figures do not necessarily Attorney Docket: 56113-0625WO1 require the particular order shown, or sequential order, to achieve desirable results. In some cases, multitasking and parallel processing may be advantageous. What is claimed is:

Claims

Attorney Docket: 56113-0625WO1 CLAIMS 1. A method performed by a quantum computer, the method comprising: obtaining an input comprising a list of ^^ quadratic Majorana operators, a predefined precision, and ^^ copies of a tensor product of a quantum state on ^^ fermion modes with the quantum state on the ^^ fermion modes; processing the input to obtain a list of expectation values, wherein each expectation value in the list corresponds to a respective quadratic Majorana operator; constructing a graph, comprising assigning a single Majorana modes included in the ^^ quadratic Majorana operators to respective vertices in the graph and adding edges between vertices that represent a pair of single Majorana modes that correspond to a non-zero expectation value in the list, wherein pairs of single Majorana modes represent respective quadratic Majorana operators; applying an edge coloring algorithm to color the graph in multiple colors, wherein edges incident on a vertex have different colors; and measuring each quadratic Majorana operator in the list of ^^ quadratic Majorana operators, comprising, for each quadratic Majorana operator, measuring multiple copies of the quantum state on ^^ modes in a simultaneous eigenbasis of all quadratic Majorana operators in the graph with a same color as the quadratic Majorana operator to obtain an empirical estimate of the quadratic Majorana operator.

2. The method of claim 1, wherein a magnitude of a difference between each empirical estimate of a quadratic Majorana operator and a trace of the quadratic Majorana operator with respect to the quantum state is less than or equal to the predefined precision.

3. The method of claim 1 or claim 2, wherein an empirical estimate of a quadratic Majorana operator estimates a trace of the quadratic Majorana operator with respect to the quantum state.

4. The method of any one of the preceding claims, wherein measuring multiple copies of the quantum state on ^^ modes in a simultaneous eigenbasis of quadratic Majorana operators with the color to obtain an empirical estimate of each quadratic Majorana operator in the listof ^^ quadratic Majorana operators comprises measuring ^^୪୭^ெ ଶൌ ^^ ^^ copies of the quantum state.Attorney Docket: 56113-0625WO1 5. The method of any one of the preceding claims, wherein the multiple colors comprise at most deg(Gε) + 1 colors, wherein deg(Gε) represents a maximum degree of any vertex in the graph.

6. The method of any one of the preceding claims, wherein processing the input comprises: measuring each copy of the tensor product in an ^^-fold Bell basis to obtain a set of measurement outcomes, wherein each measurement outcome in the set corresponds to a respective tensor product and comprises a first binary string and a second binary string of length ^^; for each quadratic Pauli operator representation of a respective quadratic Majorana operator: computing a sum of values of a variable, each value in the sum corresponding to a respective measurement outcome, wherein the variable is equal to 1 if a first binary string included in a respective measurement outcome multiplied by a binary representation of a first Pauli operator included in the quadratic Pauli operator minus a second binary string included in a respective measurement outcome multiplied by a binary representation of a second Pauli operator included in the quadratic Pauli operator is equal to 0 modulo 2 or is equal to -1 if the first binary string included in the respective measurement outcome multiplied by the binary representation of the first Pauli operator included in the quadratic Pauli operator minus the second binary string included in the respective measurement outcome multiplied by the binary representation of the second Pauli operator included in the quadratic Pauli operator is equal to 1 modulo 2; determining whether the sum of values is less than or equal to a first predetermined threshold that depends on the predefined precision; and in response to determining that the sum of values is less than or equal to the first predetermined threshold that depends on the predefined precision, outputting the value 0; or in response to determining that the sum of values is greater than the first predetermined threshold that depends on the predefined precision, outputting a square root of the sum of values.Attorney Docket: 56113-0625WO1 7. The method of claim 6, wherein the first predetermined threshold is equal to 2 / 3 of a square of the predefined precision.

8. The method of claim 6, wherein outputting the value 0 indicates that a magnitude of a trace of the corresponding quadratic Majorana operator with respect to the quantum state on ^^ fermion modes is less than or equal to the predefined precision.

9. The method of claim 6, wherein outputting the square root of the sum of values indicates that a magnitude of a difference between a positive or negative square root of the sum of values and a trace of the corresponding quadratic Majorana operator with respect to the quantum state on n fermion modes is less than or equal to a second predetermined threshold that is based on the predefined precision.

10. The method of claim 9, wherein the second predetermined threshold is equal to√ଶ ଶ√ଷ^^ where ^^ represents the predefined precision.

11. The method of any one of the preceding claims, wherein processing the input toobtain a list of expectation values requires ^^^^^ெ ^ൌ ^^ ^ఌర^ copies of the tensor product, wherein ^^ represents the predefined12. The method of any one of the preceding claims, further comprising converting one or more of the empirical estimates of the quadratic Majorana operators to respective estimates of one or more creation or annihilation operators.

13. A method performed by a quantum computer, the method comprising: obtaining an input comprising a list of ^^ k-body Majorana operators, a predefined precision, and ^^ copies of a tensor product of a quantum state on ^^ fermion modes with the quantum state on the ^^ fermion modes; processing the input to obtain a list of expectation values, wherein each expectation value in the list corresponds to a respective k-body Majorana operator; constructing a graph, comprising, for each non-zero expectation value in the list, adding a vertex to the graph that represents a k-body Majorana operator that corresponds toAttorney Docket: 56113-0625WO1 the non-zero expectation value and adding edges between vertices in the graph that represent anticommuting k-body Majorana operators; applying a vertex coloring algorithm to color the graph in multiple colors, wherein adjacent vertices have different colors and a total number of colors used is minimized; measuring each k-body Majorana operator in the list of ^^ k-body Majorana operators, comprising, for each k-body Majorana operator, measuring multiple copies of the quantum state on ^^ modes in a simultaneous eigenbasis of all k-body Majorana operators in the graph with a same color as the k-body Majorana operator to obtain an empirical estimate of the k- body Majorana operator.

14. The method of claim 13, wherein a magnitude of a difference between each empirical estimate of a k-body Majorana operator and a trace of the k-body Majorana operator with respect to the quantum state is less than or equal to the predefined precision.

15. The method of claim 13 or claim 14, wherein an empirical estimate of a k-body Majorana operator estimates a trace of the k-body Majorana operator with respect to the quantum state.

16. The method of any one of claims 13 to 15, wherein measuring multiple copies of the quantum state on ^^ modes in a simultaneous eigenbasis of k-body Majorana operators with the color to obtain an empirical estimate of each k-body Majorana operator in the list of ^^ k-body Majorana operators comprises measuring ^^୪୭^ெ ଶൌ ^^ ^ఢమ^ copies of the quantum state.

17. The method of any one of claims 13 to 16, wherein the total number of colors is less than or equal to a number of an order of ^ ఢమwhere ^^ represents the predefined precision.

18. The method of any one of claims 13 to 17, wherein processing the input comprises: measuring each copy of the tensor product in an ^^-fold Bell basis to obtain a set of measurement outcomes, wherein each measurement outcome in the set corresponds to a respective tensor product and comprises a first binary string and a second binary string of length ^^; for each quadratic Pauli operator representation of a respective k-body Majorana operator:Attorney Docket: 56113-0625WO1 computing a sum of values of a variable, each value in the sum corresponding to a respective measurement outcome, wherein the variable is equal to 1 if a first binary string included in a respective measurement outcome multiplied by a binary representation of a first Pauli operator included in the quadratic Pauli operator minus a second binary string included in a respective measurement outcome multiplied by a binary representation of a second Pauli operator included in the quadratic Pauli operator is equal to 0 modulo 2 or is equal to -1 if the first binary string included in the respective measurement outcome multiplied by the binary representation of the first Pauli operator included in the quadratic Pauli operator minus the second binary string included in the respective measurement outcome multiplied by the binary representation of the second Pauli operator included in the quadratic Pauli operator is equal to 1 modulo 2; determining whether the sum of values is less than or equal to a first predetermined threshold that depends on the predefined precision; and in response to determining that the sum of values is less than or equal to the first predetermined threshold that depends on the predefined precision, outputting the value 0; or in response to determining that the sum of values is greater than the first predetermined threshold that depends on the predefined precision, outputting a square root of the sum of values.

19. The method of claim 18, wherein the first predetermined threshold is equal to 2 / 3 of a square of the predefined precision.

20. The method of claim 18, wherein outputting the value 0 indicates that a magnitude of a trace of the corresponding k-body Majorana operator with respect to the quantum state on ^^ fermion modes is less than or equal to the predefined precision.

21. The method of claim 18, wherein outputting the square root of the sum of values indicates that a magnitude of a difference between a positive or negative square root of the sum of values and a trace of the corresponding k-body Majorana operator with respect to the quantum state on n fermion modes is less than or equal to a second predetermined threshold that is based on the predefined precision.Attorney Docket: 56113-0625WO1 22. The method of claim 21, wherein the second predetermined threshold is equal to√ଶ ଶ√ଷ^^ where ^^ represents the predefined precision.

23. The method of any one of claims 13 to 22, wherein processing the input to obtain alist of expectation values requires ^^^ ൌ ^^ ^^^^ெ ఌర^ copies of the tensor product, wherein ^^ represents the predefined precision.

24. The method of any one of claims 13 to 23, further comprising converting one or more of the empirical estimates of the k-body Majorana operators to respective estimates of one or more creation or annihilation operators.

25. A method performed by a quantum computer, the method comprising: obtaining an input comprising a list of ^^ quadratic Majorana operators, a predefined precision, and ^^ copies of a tensor product of a quantum state on ^^ fermion modes and the quantum state on the ^^ fermion modes; for each quadratic Majorana operator in the list: determining a basis to measure the quadratic Majorana operator in, comprising: processing the input to obtain a list of expectation values, wherein each expectation value in the list corresponds to a respective quadratic Majorana operator; constructing a graph, comprising assigning single Majorana modes included in the ^^ quadratic Majorana operators to respective vertices in the graph and adding edges between vertices that represent a pair of single Majorana modes that correspond to a non-zero expectation value in the list; applying an edge coloring algorithm to color the graph in multiple colors, wherein incident edges on a vertex have different colors; for a color that corresponds to the quadratic Majorana operator, determining a simultaneous eigenbasis of quadratic Majorana operators in the graph with the color; and measuring the quadratic Majorana operator in the determined basis to obtain an empirical estimate of the quadratic Majorana operator.

26. A method performed by a quantum computer, the method comprising:Attorney Docket: 56113-0625WO1 obtaining an input comprising a list of ^^ k-body Majorana operators, a predefined precision, and ^^ copies of a tensor product of a quantum state on ^^ fermion modes and the quantum state on the ^^ fermion modes; for each k-body Majorana operator in the list: determining a basis to measure the k-body Majorana operator in, comprising: processing the input to obtain a list of expectation values, wherein each expectation value in the list corresponds to a respective k-body Majorana operator; constructing a graph, comprising, for each non-zero expectation value in the list, adding a vertex to the graph that represents a k-body Majorana operator that corresponds to the non-zero expectation value and adding edges between vertices in the graph that represent anticommuting k-body Majorana operators; applying a vertex coloring algorithm to color the graph in multiple colors, wherein adjacent vertices have different colors and a total number of colors used is minimized; for a color that corresponds to the k-body Majorana operator, determining a simultaneous eigenbasis of k-body Majorana operators in the graph with the color; and measuring the k-body Majorana operator in the determined basis to obtain an empirical estimate of the k-body Majorana operator.

27. A quantum computing apparatus comprising: a quantum memory configured to store multiple copies of a quantum state on a plurality of qubits; a control and measurement system configured to operate the plurality of qubits; and a classical computing device coupled to the control and measurement system, wherein the classical computing device comprises computer-readable media having instructions stored thereon which, when executed by the classical computing device, cause the control and measurement system and classical computing device to perform operations according to the method of any one of claims 1 to 26.